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Any bullets, fire, or explosions harming the tank will negate the achievement no matter who caused the damage. Everyone has to hit the tank with only melee weapons. Therefore, you cannot have any bots, humans only. The chainsaw does NOT count as a standard melee weapon and will negate the achievement.
As for how to get it. With 3 friends, load up the Swamp Fever finale on easy. Play through making sure everyone gets a melee weapon, shouldn't be too hard to locate enough. Play through the level and first wave of the finale as usual. When the first tank spawns, put away your primary weapons and only use the melees. If there are any straggling SI, you can shoot them, but be cautious of shooting the tank. While the tank chases 1 person down, the rest should bob-n-weave to get in melee hits. It won't take long for the tank to go down. Once he's dead the achievement should unlock for everyone, regardless if they inflicted any damage on the tank or not.
I say use 3 friends because if you have a bot and kill it before the finale initiates, it will auto-spawn back and you can't friendly fire kill it on easy.