A typical tactic is to wait until the opponent has 6 cards in their hand and then you use the Annoying Dog to add the Dog Residue, giving them the 7th card. Since their hand will be full they will lose the card they draw on their next turn. You can combine this with the River Person to make sure that combo works.
The Dog Residue itself is pretty bad, making Annoying Dog even better. Due to its low stats of 1/2 but for a price of 2G and no effect, it's statistically worse than Froggit, Moldsmal and Spider, especially when unlike those, Dog Residue has no synergy with Dogs or other archetypes.