Can tasmulosin have lasting effects even after you stop taking it?

Usually not. What type of side effects are you talking about and why are you taking it? Used for prostate issues and for high blood pressure. If for prostate you may not be able to pee as well and not able to pee. Stopping it if using for HBP can result in BP going up again. So that could be called a side effect of stopping. You did not give us enough information to be of much help.

The problem that the Tamsulosin was treating may well return. Once the medicine is cleared from your system there should be no other residual effects.