Disapprove is not usually used with an object (in contrast to approve which often is).
It's because you don't take action by saying you disapprove of X, but you are expressing how you feel about X.
In business settings and similar it's common to use approve with an object, e.g. I approved time for this project, but the opposite of approve in that context is not disapprove but do not approve - I did not approve time for this project.
So you cannot omit the of.
I beg you [to do X, for X].
You are begging someone to do something.
I beg of you [to do X, for X].
You are begging someone to let you use a resource they have or control. You're not technically really directly "begging" the person you are talking to.
As a person's ability to directly help you can be considered a resource, and "doing something" can involve letting someone use a resource one has, the distinction between these doesn't matter much.
I beg of you has the connotation of sounding much more formal and polite, and somewhat archaic.