Getting Coins Into Holes In Folders Safely

Acetone (not nail polish remover, pure acetone) for the tape residue. And to push coins into folders I use a coin tube (one of the tubes to store coins) I don't know why I go to the trouble, I only put circulation sets in folders, anything UNC goes in albums for me. And yes, they're definitely snug, they "pop" when they're set properly.. Sometimes they're quite hard to get in depending on the series...

You could just use your fingers, but place a piece of SaranWrap between the coin and your finger so as not to leave any oil or fingerprint residue. Your fingers are soft, and unlike most tools should't scratch anything, unless we are talking about Proof coins.

As stated acetone should remove any tape residue fairly quickly.

If you really want to get a coin into the slots allocated for a coin, the main thing is to start by putting the ocin in on an angle. This means start at one edge of the coin and slowly push on the edge not yet into the slot. Use anything except your bare fingers. Some use a rubber malleet and just pound them into place. I suggest just using a soft cloth for this process. Using a larger coin to make the slots larger is defeating the purpose on the design of those Folders.
Folders are purposely made with the top of the slot smaller than the bottom. By that I mean the part away from the baking paper is the bottom and the top is where you put the coin in. This sort of tapered design is so that once the coin is snapped into place, it does not easily fall out. Of course worn coins are a bit smaller so fall into those slots easier. They also fall out easier. This is why so many kids wet that shinny stuff on the back of the slots with spit. That is glue left over from the manufacturing of the Folders. Wetting it reactivates the glue and the coins usually stay in place. Of course this effects the rear of the coins.
The best thing to do with those Folders is to not use them. Use an Album instead. More expensive but well worth it in the end.