Is it safe to purge the /files/styles directory? I use Image Styles to automatically generate thumbnails (for teasers and views). I have a lot of images, so I end up with a lot of cached thumbnails in the /files/styles directory.

Is it safe to delete the contents of /files/styles without doing anything like clearing the cache in Drupal? My hope is that the thumbnails would be regenerated as needed without issue.
Yes, it's perfectly safe to delete the files/styles directory: Drupal will regenerate the images the next time someone attempts to access it.

The image module defines a menu router item, $directory_path . '/styles/%image_style', that maps to the callback image_style_deliver(). 

In turn, image_style_deliver() checks to see if the generated derivative image exists: if it does, it serves it. Otherwise, it calls image_style_create_derivative().