The poison does one point of damage per second but does not need to drain the health of the target to kill them. Each time the target takes damage they stand a chance of dying (Save vs Death.) When someone is poisoned by a green slime there is no effect icon on their portrait but the character's sprite will turn green. This can be checked for best in the inventory screen.
Green slimes can be trapped with Web to prevent them from poisoning party members. They can then be killed using magic weapons or spells with minimal risk.