It's done often in hip-hop, etc...If you have individual tracks, you can put distortion on just the snare and kik or whatever it is you want distorted. But I think the few times I've heard distorted drums, even the high hats and cymbals were effected, too. Gives an industrial, machine type sound.
Personally, I would make copies of the tracks you want to distort and blend them with the undistorted tracks. Also, I would probably only do it to kick and/or snare (not overheads). Try some heavy compression (Digital Fishphones Blockfish) to make them pump, and maybe a saturation plugin, like the Digital Fishphones Dominion. Also, the SSL LMC-1 is a cool free plugin to compress/distort drums.
Route the drums mics through a guitar amp or two, and then mic that with distortion on from the amp (assuming your amp has a built in distortion channel). Could yield some interesting effects. I suppose you could also run the mics through an amp simulator.
Or record the drums to an old mulit-track cassette recorder and over-gain it, or something like that. Also could be kind of cool.
Not really sure exactly what sound you're going for, but there' also always a gated reverb. Not necessarily distortion, but you certainly not a "clean" sound.