Can you provide some context?
Hit me could be an outright demand, in other words 'punch me'. That would seem unusual, however. 'Hit me' can be used figuratively to mean you are ready to receive some news or information:
Person A: Okay, look, I have some bad news.
Person B: Hit me.
As in, 'Okay, tell me'. The use is not necessary negative, however:
Person A: I have to ask you a favour.
Person B: Okay, hit me.
Bite me is negative, impolite, and dismissive.
Person A: You can't say something like that to me.
Person B: Bite me. (as in, I really don't care for your opinion, it means nothing to me, stop talking, etc.)
The etymology of the word has connotations other than that, though. See here: http://english.stackexchange.com/questions/27753/meaning-and-usage-of-bite-me
Beg me is generally used in a situation where you have power over someone, and you want to make them plead with you not to exert this influence, or reveal whatever secret it is you know.
Person A: Please don't tell anyone about...
Person B: Beg me.