For the moment, swap out the cutting of hair for the fixing of a car, as Entisar did.
My car needs to be fixed. - Lovely.
My car needs fixing. - *I* don't love it, but I think it's acceptable in varieties of English.
My car needs fixed. - I *really* don't love it, but I've heard it. I think it's a Midwest thing (?).
My car needs to be fixing. - Wrong, wrong, wrong. Walk away. :)
So from the standpoint of grammar and analogy, that theoretically allows us:
My hair needs to be cut. - Lovely.
My hair needs cutting. - *I* don't love it, but I think it's acceptable in varieties of English.
My hair needs cut. - I *really* don't love it, nor have I heard it. Since this makes me wince and cringe, and I can't even attest its existence, *I* would avoid this if I were you.
My hair needs to be cutting. - Wrong, wrong, wrong. Should not even be on the radar.
All that said, as Entisar points out, making your hair the subject of this sentence just sounds a little unnatural in conversational English. So, back to the car:
I need to have/get my car fixed. - Lovely.
and so:
I need to have/get my hair cut. - There you go!
I need to get a haircut. - Also perfectly acceptable.
But do note, as Entisar also points out, that while:
I need to get my hair cut. and
I need to get a haircut.
*sound* like much of a muchness (and in terms of meaning, they *are*), there are in fact different underlying grammatical structures in play.