I needed a filter ring with no glass for a DIY project, so I bought a $3 UV filter from ebay and cracked the glass. Before doing so, just out of curiosity, I tested it against my "one dollar resolution chart".
These are the 100% crops from the Canon 500D that I had at the time:
with $3 UV filter:
if you stack a bunch of these you'll start seeing some effect (http://www.lensrentals.com/blog/2011/06/good-times-with-bad-filters
), but for just one filter... you're pretty safe, even if it's a shitty $3 filter from ebay
edit: other filters will definitely have an effect on sharpness; for example, cheap resin NDs, or fader NDs: http://www.similaar.com/foto/lenstests/lenstestsn.html