Will rejecting a summer program limit the chances of going to that college? For example, I apply for a summer program at Harvard and at Columbia. They both accept me for the summer program, and I decide on Columbia. Will Harvard reject me just because I decided not to go there for the summer?
No, it is highly unlikely that they would even know that you applied. They are not run by the admissions office.
@Eeyore123 Is it possible my name will show up on their system though?
There is a deep chasm between Summer students and actual matriculating students at both schools. The only thing a Summer program would show is that you were on campus and knew what the school area was like. Nothing more.