If you describe a problem or a responsibility as a burden, you mean that it causes someone a lot of difficulty, worry, or hard work.
The developing countries bear the burden of an enormous external debt...
They don't go around with the burdens of the world on their shoulders the whole time...
A burden is a heavy load that is difficult to carry.
If someone burdens you with something that is likely to worry you, for example a problem or a difficult decision, they tell you about it.
We decided not to burden him with the news.
The burden of proof is the task of proving that you are correct, for example when you have accused someone of a crime.
The burden of proof is on the prosecution.