Can sperm survive on a bath sponge?

Probably not: Probably not, and for sure not if there's even a trace of soap in the sponge. But it doesn't matter anyway: nobody will get pregnant by washing with a sponge, even with sperm present.

sperms die rapidly after they are not in a normal environment like the human body (vagina/cervix). It also depends how long they stay on the sponge. But chances are -No.