If you ask out a girl who you know likes you because of the amount of IOI's she gives you but she says she can't because she has a boyfriend even though you're pretty sure she doesn't, what do you do about this?

Women can give you indicators of interest, but it doesn't mean the comfort level is there. It just meant that she was attracted to you. Did you let her know that you were sexually interested in her? I mean BLATANTLY...?
In a woman's mind, there's no justification for a guy simply asking her out with no clear motive.
Find something about her that you find sexy and tell her.
"Wow, you like (insert what she likes). I find that pretty sexy, actually...but I'm not gonna let you seduce me tonight. You're gonna have to get to know me first. Tomorrow, I'm...(or whenever-- tell her what you're doing and invite her along)."

I hope everything works out, man.

_________________
"No novelty are you,
O' lady of all my time..."

A lot of hot babes have built up a knee-jerk reaction to automatically say no to a dates because they get asked out so much. One trick is to state that your going somewhere (with or without her) and ask if she wants to tag along.
But it may have to do with how you react to her and present yourself. You might want to study a little about how to be an Amog. A lot about being rejected has to do with your body language and how you react to the target.