The meaning of "put down the burden" is to give up a job, to shirk
The dialect of Beijing and Tianjin refers to the picker who puts down the flat burden and stops picking things. Later, it is a metaphor for abandoning the work that one should be responsible for. It has the same meaning as "to put down the pick".
It is assumed that most people encounter a lot of pick-picking behavior, when they are at school, when they are at work.
If we are in a special relationship and I am interested in maintaining the relationship and then the other party drops the ball, but I can't drop the ball, I should try to work it out to enhance the relationship. If we're in a special relationship and I don't intend to maintain the relationship, and then the other party drops the ball, then I drop the ball too, and that's a reason to go my separate ways. If we're in a partnership, and I have to maintain the relationship, and then the other party drops the ball, but I can't drop the ball, I have to work it out somehow and get the other party to accept it.