Merely failing to accept a position is not the same thing as rejecting it.So, you don't need any reasons to not be convinced of something?
So, I can reject evolution, and need not justify my position, simply because I have not been convinced?
Personally, I think you need reasons for why you reject evidence presented. Rejection is an action. It's a position. It's taking a stance about something.
First off: is this "a-heliocentrist" a geocentrist?Who decides that the burden of proof has or hasn't been met? Who decides that there is an absence of convincing arguments?
There is no impartial moderator.
What is stopping my a-heliocentrist from saying "Heliocentrists have not met their burden of proof, therefore, I need not justify my position."?
Seeing how there's an entire branch of philosophy - epistemology - devoted to answering the sorts of questions you're asking, I don't think I'll be able to sum all of it up in a quick answer in one post.
I can give one test of a burden of proof that I do think is reasonable, though: if the bar of acceptance is set so low that mutually exclusive claims all clear it, then the bar is demonstrably too low. In the case of theistic claims, this means that if, for instance, a Christian is trying to convince me that Yahweh exists and Vishnu does not, but the evidence for his case is no stronger than the evidence for the case that Vishnu does exist, then I can be satisfied that the Christian has not met his burden of proof.