The term hasn't been talked about on linguistics blogs that I know of (Mr V says, without doing a careful search), but Ben Zimmer never misses anything and he's dealt with it briefly at Double-Tongued, here. He points out the derivational connection to truther, the term for those who doubt the generally accepted view of the 9/11 attacks. In "Among the new words" (2008, vol. 83.3, p. 355), American Speech dealt with truther at some length. Here's their intro:
truther, Truther n [from truth + -er; perhaps influenced semantically by liar] Person who believes that he or she knows the truth behind some event, especially in contrast to the generally accepted explanation (compare conspiracy theorist [OEDas 1964])Anyway, a tiny question: truther clearly has to be pronounced with a voiceless [θ], parallel to the name Luther. But with birther, we have the noun birth with [θ] and the verb to birth with [ð]. (There's a set of alternations like that, including tooth/teeth vs. to teethe. A story for another day.) I think I've only heard birther pronounced with [θ], but it doesn't actually sound all that bad to me with [ð]. Surely that form's not possible, is it?
Image from TPM.