Other ways to tank the damage include using Omnitron-X with Temporal Shielding or Scholar with Flesh to Iron. Works best with 3 or 4 heroes.
Or get Haka to eat all the birdies. Or the Final Wasteland to remove them entirely (did that once - it was glorious) >:).
Others have mentioned Bunker, with his equipment and DR (possibly -2). I wanted to add that Omni Cannon works well here. Play defensively, and then take out Matty in one shot. Used this a few times.
Also, incapacitated Tempest can let you ignore the birds.
Going back to Voss for a minute; a damage race can be fun and winable with pretty "normal" heroes. Legacy, Visionary, Absolute Zero, and Tempest in their vanilla modes can get out all enemies take 2+ damage at the start of almost every hero's turn. Add Wraith with Stun for damage control and Smoke bombs with an armored hero and you can just beat Voss down. Or Ra and careful use of Twist the Ether can make heroes all but immune to a target.
Not sure if this has been mentioend yet, but Expatriette with a couple of damage buffs can insta-kill any Minions via Hairtrigger Reflexes the moment they come out - she'd need a little more buffing if a Guard comes out but Legacy and Ra can easily provide. Fight in Insula Primalis for all the damage buffs you'll ever need and laugh when Voss plays Forced Deployment :D.
Unless it's a Gene-Bound Soldier.
Not if one of the buffs is Twist the Ether.
True. Imbued Fire would help with the Soldier...but then you'd have issues with the Firesworn. But still, even if there's one type of Minion immune to the damage, you're still insta-killing all the others :D.
Only issue with Imbued Fire is that it buffs Voss's damage against the lowest-hp hero target.
That said, you could easily offset that with Flesh of the Sun God.
Hmm, didn't think of that. But if Expatriette's being buffed by Legacy as well as Ra, he can help protect/heal everyone, and if you're fighting in Insula Primalis and let al the lovely Obsidian Fields come out, I'm not sure Voss getting one single extra +1 to his damage is gonna make a huge difference that point (certainly not enough that he'll manage to out-damage all the heroes kicking his arse ;)).