No problems, except for getting vote from people who don't see him as being eligible or want anything to do with his politics.
I love the guy and think he's getting off to a tremendous start in the Senate!
But running him for POTUS will be divisive for the party that runs him, since those who like him but like the Constitution more won't vote for him and the other half of the country never would vote for him any way.
That sort of divide and conquer works against you if you want to win the WH.
You seriously overestimate how significant the birther demographic is. It's tiny and not statistically significant when it comes to actual voting. If Ted Cruz runs for President in a general election, it isn't going to make that much difference if birthers pout and stay home. Yes, every vote counts. But big picture, birtherism is a pretty fringe movement. Unless Cruz were to run in defiance of some future court decision that declared him ineligible, the extent to which people would actually vote on this issue is likely so small as to not be measurable.