I haven't seen many people filter their hard cider to get rid of bits of yeast. what will happen if I filter?
Oxidation, big time. Imagine flavors of cardboard mixed with a decadent undertone of down-pouring rain.
If you do it, do it when you pour the beer to consume it, not to package it. The off-flavors take weeks to develop, and if you're pouring it to consume it, I doubt you'll be taking weeks to drink it.
Doing a cursory google search reveals that the pore size of coffee filters* is about 20 microns. Given that yeast are in the single digit micron range (varies depending on strain, age, etc.) the yeast will pass through the coffee filter. You may retain some clumps if they've bound together on a macro-scale, but otherwise this won't work.
*I suspect that the pore size of a coffee filter is subject to brand, but this value is a reasonable and probable benchmark.
Good flocculation is the easiest way to "filter" yeast away from your product. This can be enhanced with cooling of the product, and finning agents such as isinglass (at least in beer).
Some homebrewers employ water cartridge filters when kegging to filter yeast and other particulate out of their beer. These are often available in practical 1 - 5 micron ranges at most hardware stores.