Does anyone know if the idea that something can't be proved, only disproved has a specific origin?
It was brought to prominence in modern philosophy of science by Karl Popper, who proposed falsificationism. (I cannot recommend the latter wikipedia entry though.)
I also take it that it applies to pretty much any belief, whether it's an untested hypothesis or a theory that has undergone a lot of scrutiny as to establish itself as fact, like evolution.
It doesn't apply to every belief, but only to beliefs in the form of universally quantified statements, such as
All As are X. According to Popper, real scientific hypotheses were exactly of this type. It obviously does not apply to beliefs in the form of singular existential statements, which can be verified.
Take a look at a nice blog entry about Karl Popper's typology of statements for a more detailed overview.
Lastly, Popper's idea that singular evidence has no bearing on the probability of a scientific hypothesis (known as corroboration) has been mostly rejected in today's philosophy of science. In actual science singular evidence has some kind of effect on the probability that the hypothesis is true or at least on our belief that the hypothesis is true (known as confirmation). Different applications of Bayesian accounts of confirmation are at the center of discussion in order to account for the rational structure of induction.