As you may already be inferring, today we're dealing with the philosophical question concerning the burden of proof. Who has the responsibility of proving her case? The believers or the skeptics? Which is more reasonable: to believe something until it is disproven (or weakened by reason and/or evidence), or not to believe something until it is proven (or substantiated with reason and/or evidence)?
Unless we're dealing with emotionally charged subjects, the answer seems pretty obvious: the burden of proof is always on the person making an affirmative claim. The skeptic is free to reject any claim that doesn't satisfy her standards of reason and evidence. When we are dealing with emotionally charged subjects, however, especially those on which our sense of personal identity depends, all this cool reasoning goes out the window and we tend to engage in some special pleading, unreasonably demanding that the skeptic disprove our assertions, but that's a philosophically unenlightened position to maintain, as the following thought-provoking animation by QualiaSoup demonstrates:
While the burden of proof is always on the person who makes an affirmative claim, insofar as such a claim is epistemic or ontological (and related only to whether something is likely true), the case is more difficult for moral questions, partly because these questions involve practical choices that have to be made (often immediately) rather than merely beliefs that can be held abstractly and without urgency. The difference, of course, is that with moral questions, we are not merely concerned with the question of whether some moral claim is true but also with the question of whether it is good or bad, and whether it is worth choosing and acting upon, or not...
Other difficult cases: instances involving imminent danger... but that, again, has more to do with caution and prudential choices than with mere truth.