Chris Pine is very gay friendly and has a lot of gay friends in his circle but he is straight when it comes to dating. He's seen with women very often throughout his career and he currently has a girlfriend.
As far as I can tell, there is no reason to believe he is gay or bisexual. In fact, I don't think there's that many gay rumors surrounding him.
Here are some gay quotes from Chris Pine:
“My first intimate scene onscreen was with Tom Everett Scott,” Chris Pine remembered. “He’s an incredible spooner — very warm, very sensitive.”
Chris wanted to be part of another gay movie. He wanted to be in Ryan Murphy’s (Mark Ruffalo, Julia Roberts, Matt Bomer, Jim Parsons) adaptation of Larry Kramer’s The Normal Heart.
“I would have done that in a heartbeat,” Chris Pine revealed. “Right now I just want to play good roles, and if the role happens to be a gay man, that’s not of any import other than is it a good story? Does it say something that’s interesting?”
When Zachary Quinto come out of the closet, Chris Pine said he was "so happy for him, oh man. I thought it was rad. It was really, really cool. He did it on his own time, on his own schedule."
INTERESTING FACT: Chris Pine revealed in an interview that he was in New York City for Gay Pride in 2012.
IN CONCLUSION: Chris Pine is straight is very comfortable with his sexuality and he very much supports gay rights and gay marriage!