Evidence is what gives a belief justification. There are many types of evidence, and the quality of evidence will depend both on the belief in question and what other beliefs you have. Physical evidence is only one kind of evidence. Not all evidence is repeatably verifiable, either. Some examples of non-physical evidence include the testimony of experts, direct experience and memory. All of our experience isn’t repeatable, and experience is one of the best justifications we have for our beliefs.
A proof or argument is what connects specific pieces of evidence to some claim. Arguments and proofs aren’t evidence, but they connect evidence to the claim being supported.
The “burden of proof” is merely a convention of how certain discussions happen. It is not a rule of logic or a general obligation. I much prefer the Socratic standard: When you’re in a conversation, do your best to speak truth and help find errors of the people you’re speaking with.
If you don’t accept that the person making the claim has to prove their claim, you are under no obligation to belief anything without evidence. There are many types of evidence, some of which you already have at hand that you can use to test their claim using your own reasoning.
When someone makes a claim, you can also simply withhold judgment. There is no reason that you are obligated to believe or disbelieve. However, if they are making an error, it is helpful and respectful to point out how they are making an error and why you believe that it is indeed an error.