Definition

Also called Burden of Proof fallacy, Ad Ignorantiam, Appeal to Ignorance.

This sort of "reasoning" has the following form:

• Claim c is presented by Smith and the burden of proof actually rests on side Jones.
• Jones claims that c is false because there is no proof for c.
• Conversely, Smith can claim that c is true because it has not been proven false.

This is a special case of a false dilemma, since it assumes that claim c must either be known to be true or known to be false, when it can be that it is not known at all to be true or false.

In many situations, one side has the burden of proof resting on it. This side is obligated to provide evidence for its position. The claim of the other side, the one that does not bear the burden of proof, is assumed to be true unless proven otherwise. The difficulty in such cases is determining which side, if any, the burden of proof rests on. In many cases, settling this issue can be a matter of significant debate. In some cases the burden of proof is set by the situation. For example, in democratic countries' law the burden of proof is on the prosecution. As another example, in debate the burden of proof is placed on the affirmative team. As a final example, in most cases the burden of proof rests on those who claim something exists (such as Bigfoot, psychic powers, universals, and sense data). Thus, when the claim is negative, the burden of proof is usually on those who negate the claim, since it is equivalent to asserting existence.

Examples

Since you cannot prove that God do not exist, He must exist.
Argument used by religious people.

Since you cannot prove that God exists, God does not exist.
Argument used by atheists.

Counter-examples

A map is not the territory it represents.
This claim is neither true nor false but, as a negative, the burden of proof is on the person who will claim it to be false and who therefore asserts that there is at least one case where a map is the territory it represents.
Such is the case for every premise of a scientific theory, considering that the premise will hold on every presently known case. As soon as one counter-example is shown, the premise is known as false. So far, no premise of a scientific theory has ever been known as true.

Advices

No real case of argument from ignorance can be used to prove anything. You can only state some claim as a premise. Just be sure that no counter-example has been found.
If your opponent uses such argument, tries to reject the burden of proof on you, point out that you are not on the affirmative team or exhibit a counter-example.