This rather common verb phrase is used referring to an angry person overreacting to a trifling matter, specifically to something that someone has said or done.
The last instance of "bite someone's head off" I came across was recently (late 2006), in the third season of an exhilarating Warner Bros. sitcom created by Chuck Lorre, “Two and a half Men”, which I have the pleasure to subtitle in French for the channel Canal Plus. In the ninth episode, entitled “Madame and her special friend”, the uptight Alan Harper (starred by the outrageously funny John Cryer) shouts in frustration at his whimsical brother Charlie (both a homonymous clone of and an eponymous role for Charlie Sheen). The reason for this fit of anger is that Charlie does not keep his promise to “give it a rest” on the jokes his makes about Norma (starred by Chloris Leachman, who needs not be introduced), Alan’s elderly date (hence the title). Charlie, judging that his sibling makes yet again a mountain out of an anthill, retorts:
“You don’t have to bite my head off !”.
Wether in or out of this context, the exaggerated therefore comical catch phrase may be thus paraphrased: “I do not deserve such a harsh punishment for such a petty mistake !”. I hope this explanation will be of any help to you.