Because everyone and their grandmother is doing it, I may as well write up the Doctor's feelings on sex and relationships. Keep in mind that this is only my interpretation.
Overview
Short answer, he's not interested.
Okay, okay, let's go more in-depth. Relationships, in his mind, are a bit like staying in one place. They root you to something, someone, and make all that running and danger and inevitable leaving-behind that he does a lot more difficult. They shift the focus from saving planets to what's for dinner. It's boring and it's domestic and he simply isn't patient enough for it. Besides that, he has experienced a huge loss. Just taking companions, let alone forging a deeper connection with them, sets him up for more of that.
Sex is different. Sex, for the Doctor, is like ping-pong. It's fun for five minutes, but there are much better uses of his time. He doesn't really have a sex drive, and when he flirts it's either out of reflex with no real thought behind it or as simple joking around to be abandoned as soon as the other party gets serious. He has no real preference for gender or species (though intelligence is a must), but good luck getting him interested.
The Master
Since Nine has no Master to call his own, these observations are based on RP.
This Doctor has a lot of pent-up rage and the Master really brings that out in him. While they can and will engage in playful banter, it can escalate to threats and violence at the drop of a hat. As such, this makes any kind of romantic or sexual relationship, at best, difficult. Besides, he doesn't see the Master that way. Except on rare occasions, he doesn't even see his childhood friend.
Romana
(Ditto that.)
Since she is dead in his time, Nine views Romana with the fuzzy glow of a memory made fond through time and loss. He's nicer to her, more willing to take what she dishes out, though still plenty capable of snarking back. They're friends, certainly, but something more? Unlikely.
TARDIS
Nine loves his ship. ♥ He really, really does. She is his constant companion, the only one who can or will stay with him forever. That doesn't mean he's plugging anything in where it doesn't belong, but he has been known to caress her console and that carries over to her projections on the astral plane.
Rose
Canon: Despite what some of the fandom says, I don't believe Rose is his Tru Wuv. I don't even think he loves her more than any of his other companions. The only thing that makes her stand out is that she was present at a very needy time in his life. He gets jealous when she flirts with other guys because he doesn't want to be left alone again. Anyway, he has no problem hugging her and dancing with her, but he really doesn't see her as more than a friend.
realityshifted: He likes her, really he does. He just doesn't understand why his future regenerations are so batshit over her. She's a lovely girl, but really.
Mickey
Canon: Idiot. Somewhat useful idiot, but still an idiot.
realityshifted: They have a great Doctor/companion bond and they high-five and hug and all that when they save the world. The Doctor might even plant one on him if he was feeling particularly fantastic, but it would be a fairly platonic kiss. (Note: The Doctor has no problem just snogging people if the mood strikes, but he doesn't intend for it to be seen as a sexual advance. There's a difference between a smack on the lips out of sheer joy and making out, after all.) Also, they have been known to watch alien porn for a laugh.
Jack
The Doctor only flirts with Jack because he will never, ever give in and so it is hilarious to watch Jack get his hopes up.
Jackie
No.
Self
Obviously, the Doctor loves himself. However, he much prefers bickering with himself to sexing up himself, so it is unlikely to ever happen.