Holes are most often made in polymer clay before hardening, but they can also be made after hardening with drill bits in electric drills, or even by hand.
I usually keep my Dremel in a drill press and just use the size bit I need for the hole I want (sometimes it's hard to find the smallest diameter drill bits). I've never noticed a problem with friction/heat, perhaps because the holes are drilled so fast and/or most Dremel speeds aren't super-super fast?
Polymer clay is soft enough that it can also be drilled by hand after hardening. That's usually done with a drill bit that's been embedded in a polymer clay handle, perhaps after making a pilot dimple. There are also various mechanical drills like pin vise drills, push drills, finger drills, etc, that can work.
Holes can also be burned through cured polymer clay with a red-hot needle held in a pair of pliers, but the clay that's burned will be extruded as a tiny coil at the top of the hole (this is usually done to make a short hole for inserting the shank of an eye pin in a dangle if not done before curing) along with a puff of smoke--not great to breathe.
You can read about all those things on these pages of my site if you're interested:
http://glassattic.com/polymer/tools_Dremels_worksurfaces.htm
...Dremels & Regular Drills
...Handles
(example of one of my drill bits with polymer handle, on right:
http://i96.photobucket.com/albums/l163/DianeBB/tools/e751.jpg )
http://glassattic.com/polymer/beads-holes.htm
...Holes After Baking
http://glassattic.com/polymer/pendants_cording.htm
...Flat Holes
Muslin buffing wheels (often with stitches removed) or homemade fluffy buffing wheels are used by many clayers in Dremels or other rotary tools for small buffing jobs. (Larger jeweler's buffing wheels and garage buffers are also used outfitted with muslin or cotton wheels).
Other bits can be used for polymer clay too but aren't used that often:
...the solid felt buffer bits can be used for shaping and grinding polymer clay
...cutters, engravers, etc
...however, some clayers use various Dremel bits or shanks to make their own "sanders" or "buffers" for the Dremel too (couple of examples:
http://www.desiredcreations.com/howTo_TLDremelPolish.htm )
As for drilling in resin, just use a regular drill bit but be sure and have a piece of wood underneath the resin piece. And if you're doing more than a little (or sanding resin), wear an appropriate mask.
(Lots of people use glue-on bails, etc, instead of drilling holes though.)
Much more info here:
https://www.google.com/search?q=how+to+drill+resin
.