Hi, Lori. There is no Burnham county or town called Burnham that I can find online.I know some people involved with the pioneer heritage group in Omaha and there's some experts involved with Mormon Trail Center. The next time I'm down there, I'll inquire around. If I find a location out, I'll post it.
What is your ancestor's full name? That could prove helpful as well, if anyone else has documented something about him.