It feels natural to me. "Keep" is used in the sense of "observe", and "hygiene" is used in the sense of "good health practices".

The grammar is fine, but it's not natural at all and it doesn't even mean what they obviously intended it to mean. You might remind someone to "mind" their hygiene or to "remember" their hygiene, but that has nothing to do with where you put your used toilet paper. I would suggest: Please don't flush your used toilet paper. Put it in the trash.