Burn the Impure deals 3 damage to target creature. If that creature has
infectInfect is a static ability.
Damage dealt to a player by a source with infect doesn’t cause that player to lose life. Rather, it causes that source’s controller to give the player that many poison counters. See rule 120.3.
Damage dealt to a creature by a source with infect isn’t marked on that creature. Rather, it causes that source’s controller to put that many -1/-1 counters on that creature. See rule 120.3.
If an object is no longer in the zone it’s expected to be in as an effect causes it to deal damage, its last known information is used to determine whether it had infect.
The infect rules function no matter what zone an object with infect deals damage from.
Multiple instances of infect on the same object are redundant. (CR 702.90), Burn the Impure deals 3 damage to that creature's controller.