Final answer:
The Common Cold (ItemID: 103) is the item in 'The Binding of Isaac: Rebirth' that grants tears the chance to poison enemies, with effectiveness scaling with the player's luck stat.
Step-by-step explanation:
In "The Binding of Isaac: Rebirth," the item that provides tears with a random chance to apply a poison effect, inflicting 4 or 6 damage per tick to enemies, is A) The Common Cold - ItemID: 103 "Poison damage". This item's effect of poisoning enemies is indeed influenced by the player's luck stat, with higher luck increasing the chance of this effect occurring. At +12 Luck, the poison effect will trigger on every tear.
Furthermore, when combined with the Dr. Fetus item, the poison damage from The Common Cold significantly enhances the damage caused by the bombs. This synergy makes it an effective combination against enemies and bosses in the game.s