BEFORE GOING INTO THE ITEMS READ THIS
This season 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. With that being said, I have put the breakdown of every item I have on the build option section.
Iceborn Gauntlet: Even after the item rework it is still a 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 pottential 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 since the mythic system is gone you can now 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. It also works with Ingenious Hunter to be able to proc it quicker. If you were to proc the item on cooldown you would be able to have 3 uses within a minute instead of 2.
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 ability power 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, on top of the passive given from Deominc Embrace, so even more ap when building tanky.
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.
Black Cleaver: It's a solid choice when looking to shred down tanks this season. Even though the shred has been nerfed, it's better now, as tank items give more defence and make it harder to kill them normally without it.
Sundered Sky: Very good item on Volibear. The sustain is nuts and fills the gap Divine Sunderer left nicely, if not even better. 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. With the changes it got this patch it's even better for you. You don't mind losing 10 ad for 150 health, you actually like it more this way as it allows you to sustain even better. A lot of champions currently 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: With the extra health is got it can actually be done as a rush item to gain a very large amount of sustain in lane. At level 9 with a Doran's Ring starter item you just need to proc Grasp of the Undying 5 times to get the effect going, which is very early and basically gives a mega Garen passive.
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: The changes it got in season 14 made it an anti mage item once again, very good once fully stacked, giving a ton of resists and speed.
Overlord's Bloodmail: It can provide you a lot of extra damage as a 3rd item option to give you over 80 attack damage when you end up buying it.
Kaenic Rookern: The best anti mage item in the entie game right now. 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 attack damage is rather weak compared to other juggernauts. The shield however is good and you can increase it when you use your ultimate.
Stridebreaker: With the removal of the mythic system, Stridebreaker is not a much more viable option. 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.
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. However, if you have Ingenious Hunter, combined 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 ability power 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.
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, but it's been nerfed severely. It's only used to heavily split push now, the problem is it doesn't give enough to make it worth buying. You are buying hullbreaker to get a demolish proc every 5 hits, and most of the time it's not good enough.
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: Since the cooldown got cut to 5 seconds from 7 it can now be built reliably, as it's basically like having almost all 5 Ingenious Hunter stacks right off the bat. You can still do the 3 core tank setup well.
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 ability power 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 Naure, 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, possibl 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.
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. The cost reduction is also a massive thing. It is a very cheap item now, and will make it very frustrating for auto attack based champs to play out the lane.
Nashor's Tooth: Very good item for an ability power focused build. It will make your attack speed insanely good, but it's best used into a full ability power build. As a standalone item into a regular build it's not good enough.
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.
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 on the 1st hit 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.
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.