What do you believe is true even though you cannot prove it?

That things continue to exist when not being observed. Since evidence requires observation, no evidence could be provided for this claim, but I think it is true.

Few reasonable inquiries use proof as their standard, but rather evidence is called for. I try to believe the truth, and so tend to support what the evidence shows. Belief really isn't a vary useful word for this sort of thinking.

That the pun is the greatest form of humour known to the English language:

Otis: It's all right, that's in every contract. That's what they call a sanity clause.

Fiorello: Hahaha! You can't fool me. There ain't no San-ee-ty Claus!

that houses can be haunted (not by ghosts). By a 'something' which cannot be understood from a logical point of view.

that I have another bottle of scotch around here somewhere.

I'll get around to proving it before midnight.

You proved how truly dumb your question is.

You and I are related.

That I am conscious.

Jesus was false and homosexual