wipe someone or something (off) (with something)
to clean someone or something by wiping with something. Tony wiped the baby off with a soft cloth. Jane wiped off the counter with a rag. Tom fell in the mud and asked Ralph to wipe him off.
wipe something (off) (on something)
to remove something by wiping it on something else. Don't wipe your feet off on the carpet. Don't wipe off your feet on the carpet, wipe them on the mat.