Most pass plays have the HB blocking or going to the outside for a dump (in which case he gets shadowed by the MLB and easily tackled). The dump passes are either designated short passes, or medium passes where the QB doesn't see anyone open and has the dump pass option on.
The only play where I've seen HBs really go downfield consistently is on designated long passes. Here's an example of a HB making a deep reception, and it was on a long pass. I honestly don't know why the QB thought he was most eligible, seeing as he was running through two defenders and has less than 20 catching. But hey, it worked
http://goallineblitz.com/game/replay.pl?game_id=121117&pbp_id=5620469