To me, this is ungrammatical. For me, the pattern goes like this, where the asterisk next to (4) means it's ungrammatical:
- Answer some emails
- Respond to some emails
- Answer the fuck out of some emails
- *Respond to the fuck out of some emails
[answer [some emails]]
[respond [to [some emails]]]If we add in the emphatic the fuck out of, then the fuck becomes the object of the verb, and we get something more like this:
[answer [the fuck] [out of [some emails]]]It should be understood as meaning that the fuck, whatever it is, is the thing that is removed from the emails due to their being answered so comprehensively.
So if, like for the tweeter, it's grammatical with respond as well, then we must have an identical construction: the fuck is the complement of the preposition to, and we've got this:
[[respond [to the fuck]] out of some emails]But for me, respond to the fuck doesn't work so well. It doesn't even have the idiomatic meaning of 'doing something so well it removes all the fuck from it', which we saw above. out of some emails has to be an adverbial phrase, I think, being something the location of the whole action. You'd have to respond the fuck out of some emails, and in fact that is much more acceptable to me.