The implication of "the problem" or "the problems" in "I don't want to be involved in the problem(s) of my boss."

1 implies that the boss has problems of his own that you do not want to be involved in.

2 is more ambiguous. It probably means that your boss is himself or herself a problem of some sort (bad manager, incompetent etc).

2 is less likely to mean that your boss has only one problem and that you do not want to be involved in it. This meaning would normally be expressed by something like “I do not want to be involved in my boss's problem”