Line the holes with felt? Or do you mean line the side(s) of a cylinder? Or put felt only at the bottom of the hole? A spray adhesive, with a straw-tube (like the tube WD-40 has) might work.
How thick of felt? Felt on a pool table is as thin as upholstery fabric, but felt for upper padding of coil springs (upholstered chair seats, etc.) is 1/4" thick. How thick of felt are you wanting to apply? I assume the felt is thin, like pool table felt. I know of spray adhesive for upholstery applications. Maybe a local pool table shop (repair shop) might have some good ideas for thin felt.
For the bottom of the hole, maybe the felt pads for furniture leg/ feet, with the peel and stick adhesive, will help. Wal-Mart or any similar outlet has these peel and stick felt pads.
For inserting felt into a glue lined hole, wrap the felt around an appropriate sized dowel rod and jam it in.
If you are wanting to line a hole for tool rests or some similar object that needs to rest in a cushioned/lined hole, maybe your best bet is to cut through the holes/cylinders, ie., 2 halves, line the halves with felt, then re-attach the halves to make the whole "holding unit".
Sonny