My employer has started requiring we put used toilet paper in an open trashcan bin in the bathroom. What are the health effects?

I'm guessing you're American based on how you write (calling it a store instead of a shop, for example). I've linked an OSHA page. Your employer has to provide access to a certain number of flushable toilets (depending on number of employees). Check out the link, the specific definition of interest is "water closet", but read the other definitions too, then hop down to section 1910.141(c)(1)(i).

You don't mess with OSHA, they can put a financial hurt on a company very quickly. That should be reason enough alone to convince him to fix the toilet (which is probably a really cheap issue to deal with anyway).

Urine and fecal matter are health hazards.

Additionally, whose job is it to take out the trash at your office? I'm going to guess that this was not described anywhere in the job duties nor discussed when you interviewed for this job, right? Probably the same for your coworkers. If it isn't your job, you can't be required to do it.

In the US that's a health hazard enough to be a legal problem. Someone up the line has to provide the building with working toilets and trash pickup or else! You simply call the city hall and have them send you to the health board or whoever.