"Do you have any problem "generally, is considered rude but it all depends on the context and intonation of the speaker, a rather correct way to frame it interrogatively would be "Are you facing any problem" ?

No, it need not be offensive. In certain situations it can be an offer to help. For example, if you see a motorist by the side of the road struggling with his car you might ask him “do you have some problem?”

Also when a company or organization has a staff meeting or border meeting where many decisions should be made, let us say some people agree on a solution the manager or head of the meeting can ask; Anyone have a promblem/ objection with *bla bla* ? Today i was in a meeting and the manager of this organization used this ALOT after we went through each issue that we needed to solve. So it can also mean generally if you have anything against something. Yes it can be rude as well for instance; *loud people on the bus* and you say; hey you are bit loud! Well then they can turn to you; So do you have any problem with that! ( which basically is a common way to start a fight )