The Burden of Proof isn't an absolute property. There's no experiment you can perform on a person, a brain, or a position to show that that thing has the burden of proof.
The Burden of Proof is a social construct. Consider that in physics, there's a concept of inertia: objects at rest stay at rest, objects in motion stay in motion. In the realm of ideas and beliefs, the same concept is there: we have inertia with respect to our ideas. Just like it takes work to change the motion of an object, it takes work to change the beliefs of a person or of a group of people.
The Burden of Proof is a way of expressing the need for that work. If you want me to change my beliefs, you must put in some work to overcome my inertia.
If you don't have any desire to change someone's beliefs, then you also have no burden.