My desk top computer is set up in my living room currently with a regular rolling office chair. My floors are ceramic tiles. The chair is about 2 years old. Recently something appears to be coming off the wheels and leaving black sticky marks on my tiles. I do not want to place any type of mat under this chair as it's in my living room. Now, I am constantly cleaning those marks on the tiles. Any suggestions to put something on those wheels to prevent this?
Any info would be helpful. Thanks.

If you can't clean the casters and don't want to put down a mat then pop the casters out using a flat blade screw driver and go to the hardware store for replacements. Choose ones with a different surface material than what you have now.