None of the Toy Story films mention Andy's father.

After three feature-length films without even a glimpse or a mention of a father figure, it is fairly obvious that Andy's mom is a single mother. When asked about the situation, Toy Story 3 director Lee Unkrich said, "We don't mean to be mysterious about it; it's just never been relevant to the story."

Unkrich goes on to point out the fact that a 'complete' family can seem a bit too perfect when making a film. Having a character with a missing parent provides many opportunities for emotional moments that can connect with audiences on a very personal level. The screenwriters and directors have declined to comment on the whereabouts of Andy's father, but Unkrich did speculate a bit on the origins of Woody the cowboy:

"Maybe Woody is so special to Andy because he was given to him by his dad. I don't know if that is the case or not, but there is that kind of unspoken question, 'Why would Woody be especially special to him?'"

Perhaps we will find out in a Toy Story 4...
(source)