Some things may remain unproven and unresolvable. I don't think it really matters much if people believe it or disbelieve it, at least with most things. It only really seems to matter when it comes to the realm of public policy and national politics.
It's also important in the legal system, as proof beyond a reasonable doubt is the standard which they're supposed to follow before convicting someone of a crime. In those cases, if you can't prove it, then by law, they're not allowed to believe you. It's an adversarial system, so if one side does sloppy investigation or if their dog ate their homework, then they may still end up losing the case, regardless of what the actual "truth" might be. "Proof" can sometimes be far too subjective, which is how innocent people can be sentenced for crimes they didn't commit. Or sometimes, the guilty might go free.
In the science world, it's a bit different, at least inasmuch as no one is in danger of going to prison if someone can't prove their claims. The worst that might happen is that someone might get ridiculed and mocked, which is how mad scientists are created. ("They laughed and made jokes, but now I'm going to prove them all wrong and take over the world! Bwahahahaha!")