This fallacy originates from the Latin phrase "onus probandi incumbit ei qui dicit, non ei qui negat"). The burden of proof is on the person who makes the claim, not on the person who denies (or questions) the claim. The fallacy of the Burden of Proof occurs when someone who is making a claim, puts the burden of proof on another party to disprove what they are claiming.
Ellis: "I believe that God exists." Marty: "How can you prove it? Ellis: "I don't have to, if you can't prove that He doesn't."