Reasons to think Ike likes girls
Calls Elincia a "good luck charm", defends her fiercely from the apostle, holds her hand, and is sad if she dies. Shinon and Ranulf make fun of him plausibly being in love with her.
Has in-game dialogue and supports with girls he becomes friendlier with like Elincia and Lethe.
Priam exists and is Ike's descendant. Probably. He and the flavor text say so but there certainly is some non-committal writing.
In Smash Bros, there's one clip where it looks like he's giving a flower to Palutena.
Reasons to think Ike doesn't like girls
Is extremely off-put by Aimee, but to be fair, she is acting like a creep.
Despite becoming friends with Elincia and Lethe and working with Mia for years, he can end up with none of them in the end and explicitly has no desire to pursue any relationship with Elincia even at the end of Path of Radiance.
In fact, he has no support endings with women at all, though he does only have two total.
He was so disinterested in the temple handmaidens, he just walked away without a word. He prefers the pretty flowers he ignores to do his job, which he prioritizes over absolutely everything except Soren's happiness. To be fair, Ike only prioritized making Soren happy over doing his job twice... Oh wait, it was three times but the second was unsuccessful because it was quickly interrupted.
Aimee is convinced if she didn't tell Ike she liked him, he wouldn't know. He tells her he wouldn't do anything about it even if he did know, which can be extrapolated to explain his non-reactions to Elincia and Marcia's potential flirting.
Aimee hypothesizes that Ike isn't interested in settling down yet, and Ike seems to believe she is wrong... He says she is delusional a few lines later, but to be fair, that was after she suggested she may have a chance with him in the future.
Calill, Goldmary, and arguably Hortensia try to get him to acknowledge their attractiveness. It doesn't work.
The art book explicitly states that he doesn't. Only, that's just supplemental material and a different translation of the same phrase could also mean that he doesn't notice their affections (see point 5).
These were the ones I could think of. If there are any others from Heroes, Engage, 3DS DLC, or other supplemental material (or even on the Tellius games that I forgot about), let me know!
Decide for yourself which is more likely!
Reasons to think Ike likes boys:
I can't list them all right now there's too many
Reasons to think Ike doesn't like boys: