Removing Prints From Coins

Not really. Any rubbing with a soft cloth can be seen on a shiny coin. Since it is circulated, it won't hurt it too bad. For circulated coins, I sometimes use those silicone lens cleaner cloths I get for my glasses. Don't touch proof coins with those.

You can also try acetone. Just hit the search button to find a lot of articles about it. It's one of the very few 'safe' things to do to a coin.

I had a Nickel once with a well defined fingerprint on the reverse. Not really worth much so I tried just about everything I could find. My end result was with a butane torch. That worked but naturally the coin had some nasty after effects.
As noted try the Acetone Method. I not to long ago purchased a 1995 Double Die Lincoln Cent for a song due to a fingerprint right on the middle of the obverse. It was at least MS65 so worth the attempt to get rid of the print. Must have been recent since it mostly came off after several soakings in Acetone. Still not happy and with not much to loose I tried fingernail polish remover. That finished the job. Print now all gone. Coin looks great.
I do not recommend the usage of fingernail polish remover since there are numerous manufacturers with many, many different formulae for that product and what worked for me may destroy someone else's coin.