BEFORE GOING INTO THE ITEMS READ THIS
Volibear has an absurd amount of diversity. The synergy he has with some of the new items is very good, and some of the old item changes are actually better for him. I highly recommend testing and experimenting to see what playstyle you like best. I have put all the items I consider viable, but there might be some you like more. I have tested almost every single item at this point, with some exeptions like not taking any crit items on Volibear, it's his worst stat by far and I think it's never proper to build a single crit item on him, exept Navori Flickerblade, but you never buy it because of the crit. With that being said, I have put the breakdown of every item I have on the build option section.
Iceborn Gauntlet: Very consistent item on Volibear. In terms of Sheen items this one is the best generally. It provides health, a reliable slow and good armor. Your stick potential is very good once you complete it.
Heartsteel: Any %health damage will absolutely anihilate you when building this. With that being said, it can still work if you want to go for a heavy tanking build. It gives massive health late game and you can pair it up with Jak'Sho and Iceborn Gauntlet. It's much better on him now and can work very well as a standalone item as well if the enemy doesn't have %health damage or doesn't normally buy %health damage in items.
Jak'Sho, The Protean: It is a very solid choice. Not a whole lot to say here. It can be built into mixed damage matchups like Akali or Jax, or can be solid for trying to build very tanky and just being a meat shield for your team.
Trinity Force: This item had very little time when it was actually the best choice for Volibear. Right now it feels very lackluster compared to the other items, especially seeing as it's lost the threefold strike passive as well. The big Sheen procs is what the item is most known for, but Volibear's base damage is very low and would not make it all that great. It doesn't feel terrible, but it's not very good either. If you are snowballing out of your mind perhaps you can, but even then I would rather buy Iceborn Gauntlet, or if you really want to you can pair it up with Spear of Shojin and explode people.
Riftmaker: Very good item if you want to go for a more AP focused build. On top of that the omnivamp is amazing on Volibear. They changed how it works and now is even better for him, as his ult and E are AOE, so now that he fully heals off it, it's great.
Cosmic Drive: Not as good as it used to be, in part due to the fact that the new bruiser items and bruiser item changes make Volibear very happy to go for those. That being said, it's still a solid choice for speed, damage and getting ability rotations out much quicker. The only issue is that there's a lot of matchups where this item was very good to buy, but now are almost always bad to buy into, one example is Aatrox. Him getting Sundered Sky now means you would much rather want to sustain in melee instead of quickly running in and out of fights with short trades.
Rod of Ages: At first I didn't really like it all that much. There is a build setup where you do this item plus Navori Flickerblade, but I have been steadily warming up to it, and it feels quite good. It is a very good item when doing a specific 3 core setup with this, Navori Flickerblade and Hullbreaker. You becomes a split push MONSTER.
Black Cleaver: It's a solid choice when looking to shred down tanks such as Ornn, Rammus, or other hyper armor stackers.
Sundered Sky: Very good item on Volibear. It's very good in 2v1s or 3v1 as you can proc it very quickly multiple times, and in lane it makes it so you almost always win every single short trade, especially against tanks. A lot of champions buy Sundered Sky, and if you are in one of those lanes you can also just match it and you also buy it and do really well.
Warmog's Armor: It can actually be done as a rush item to gain a very large amount of sustain in lane. With a Doran's Ring starter item you just need to get an extra Ruby Crystal, or a bunch of Grasp procs and you have a mega sustain tool in lane.
Thornmail: Solid heal reduction item. Not a whole lot to say here. Works well against a lot of healing and if you need defensive stats.
Force of Nature: Very good once fully stacked, giving a ton of magic resist and speed.
Overlord's Bloodmail: It can provide you a lot of extra damage as a 3rd item option to give you over 80 AD when you end up buying it.
Kaenic Rookern: The best anti mage item in the entie game for tanks and bruisers right now, in most cases. If you struggle to survive certain AP matchups in lane just rush this and they won't ever beat you. It's hilarious how strong it is. You will always come out on top of short trades, and basically eat any chip damage they deal.
Guardian Angel: Can be a very good last item for a specific teamfight to end the game. Usuaully late game a single fight will decide if the game ends in a win or a loss, so if you have the revive for it at that point then it's great. Can also sell your full boots to have it in the hyper late game.
Sterak's Gage: Volibear won't get a lot of attack damage from it as his base AD is rather weak compared to other juggernauts. The shield however is good and you can increase it when you use your ultimate.
Stridebreaker: It's not anything too amazing, but the stats are decent and the slow effect is quite good as well. Not a super amazing item, but it's decent. If you use the active of the item while your are in your Q the animation will play as if you actually used the empowered attack to stun someone, possibly confusing the enemy. This works great vs Fiora as the will most likely parry thinking you used the attack to stun.
Titanic Hydra: The active is nice on Volibear, but it's quite hard to fit into the build regularly. The issue is that he doesn't have enough room to buy everything he wants, so he would have to forego Spear of Shojin, or Sundered Sky in favor of it. Even though it's not bad the tradeoff is meh in my opinion. That being said, can still do a lot of damage and work.
Manamune: Insanely good damage item. The sheer damage this item gives for Volibear is nuts. Also, because it is a Tear item it does mean your mana issues will feel a lot less punishing.
Winter's Approach: Not the best defensive item, but it does get the job done. If you don't need the damage from Manamune while also needing more mana it feels decent enough. If you combine it with Spirit Visage and Unending Despair you will be very tanky. That 3 core is very strong for just tanking through a lot of damage.
Archangel's Staff: The weakest out of the 3 Tear items, but it's not bad. It can be bought if you are going for an AP focused build, and the shield makes you a bit tankier.
Edge of Night: This item in particular is very good against Vayne. She can never push you away when you have it. Other than that there's not much use for it. It hard counters Vayne, and that's about it really. You can also try it into Singed as he also can never Fling you off him.
Serpent's Fang: Good item against a team comp with a lot of shields. Your E will also proc it so you can do it from range at a safer distance if you wish to.
Chempunk Chainsword: The offensive heal reduction item for Volibear. Great against a lot of healing where you need to proc it very quickly on priority targets that you dive on, like Soraka.
Hullbreaker: Volibear uses it rather well. It's only used to heavily split push now, the problem is it doesn't give enough to make it worth buying most of the time. You are buying Hullbreaker to get a Demolish proc every 5 hits, and most of the time it's not good enough. The time it's actually good is when you take it in combination with Rod of Ages and Navori Flickerblade. Due to the reduction from Navori you can get A LOT of procs very well.
Navori Flickerblade: This is the second item you will be doing in a 3 core setup consiting of Rod of Ages, this item, and Hullbreaker. You get a lot of reduction on your spells due to the fact that both your Q and W proc the effect on top of your auto attacks, and both abilities are also auto attack resets, so they work very well.
Spear of Shojin: It has become an incredibly good and underrated Volibear item. Every single one of his abilities gets the damage amplifier from it, including his Q. He will stack it very quick and have his W deal the maximum of 12% damage every time. The Ability Haste is also very good for him.
Unending Despair: You can still do the 3 core tank setup well with it and be very tough to beat, especially vs Jax, as I have found it to be the only way to ever stand a chance vs him.
Spirit Visage: Spirit Visage has now become a lot more reliable to build on Volibear due to the fact that you can pair it with Sundered Sky to get even more heal amp. The increase in shielding and healing is very good if you have more than your own from enchanters like Soraka, Lulu, Taric. The problem is it still does not give you enough magic resist against AP users most of the time. However, it can be used as a softer mr choice if the enemy champions have a lot of mixed damage, like with Akshan, Irelia, Jax etc. These champs have some magic damage in their kit that isn't insane, but the mr does help into it. If you need heavy magic resist though, take Kaenic Rookern or Force of Nature, much better.
Abyssal Mask: Very niche item. Can be used as a decent shred to magic resist when you use your ultimate to get into the middle of a teamfight, but it's basically never bought as the other mr options for Volibear are much better and much more reliable.
Dead Man's Plate: It has to be bought as a 4th or 5th item. It doesn't feel amazing for Volibear in top lane. You lose out on other more important core items, and it's more so bought if you want to be extrmely speedy and pair it up with other items, possibly a hit and run playstyle paired up with Hullbreaker, where you push waves to get close to the turret, hit it 5 times and then run away to safety because of the extra slow resist on it in case people want to collapse on you. You can also sell your boots for it and buy it for tankiness and speed in the hyper late game.
Randuin's Omen: Solid tank item. Works best against champions with crit. The active slow also works nicely when jumping in the middle of a teamfight and using it.
Frozen Heart: Very good rush item into certain champion matchups like Tryndamere or Yone. It also fixes a big part of your mana problems during the laning phase. It is also a cheap item, and will make it very frustrating for auto attack based champs to play out the lane.
Nashor's Tooth: Very good item for an AP focused build and split push build. Your attack speed will be insane with it.
Stormsurge: Only ever built into full AP burst builds, it doesn't really have a place in other ones. It makes it so if you one tap someone in a teamfight quickly with an ult hit the enemies also take a massive aoe from it as well.
Rabadon's Deathcap: It is best used into an ability power focused build. It will spike your damage massively when you complete it later on.
Lich Bane: It is best used into an ability power focused build, or to sell boots and buy it for the speed and damage in the hyper late game.
Zhonya's Hourglass: This can be the last item in your build, or can work into an ability power focused build. The stasis is amazing at buying time, and if you use your E then use the active, by the time you get out, you will have the shield on you.
Mejai's Soulstealer: It is best used into an ability power focused build. Best bought if you think you can hard snowball a game. You can sit on Dark Seal for most of the game if you want though and not upgrade it until much later
Blade of the Ruined King: The item in theory is very good for Volibear. Both his Q and W proc it, but sadly the lack of resistances is too glaring. Unlike an item like Cosmic Drive, it doesn't give any speed, aside from the passive that doesn't even last that long, is more expensive, and the stats it gives are less valuable on Volibear at that particular stage of the game. It is best bought as a 4th or 5th item in very niche scenarios, like vs hyper scaling hp tanks like Sion, Cho'Gath, etc.
Wit's End: Nashor's Tooth is a better alternative to this most of the time. It is not a bad item, and can be used decently well, but it fits very awkardly into most builds. It is however viable in very niche scenarios.