As a purely academic point, I believe best practice is to put the lightly worn F tire on the rear and the new tire on the front. However, if you have a tire in good condition with little visible wear on the front, you are more likely to screw something up (pinched tube, displaced rim tape, damaged valve stem) by doing the swap, than you would be to have the tire fail because it is arbitrarily 'older' than the rear tire. So the pragmatist in me says, unless there is some reason you want to replace the front tire, leave it as-is under the doctrine of 'never fix a running part', and just replace the rear.

Also, as others have said, as long as the tires are generally the same size and type it will be fine. Even a couple mm difference in width (like 25mm to 28mm or so) is not a big deal, but having, say, a 38mm tire on the rear and 23mm tire on the front will likely change how the bike feels and handles - but even that you would likely get used to it within a couple rides. Also, if you have a road slick with no or minimal tread on one wheel, it might not be ideal to put a rugged knobby tire on the other wheel. Again, it will likely not be unrideable, but it might feel a little funny til you get used to it.