Is the noun "liar" considered offensive? Would it be offensive to use it with a proven liar?

Calling someone a liar is making a very aggressive statement. If you do so in a bar, don't be surprised if you get a beer bottle upside the head. If you call someone a liar in person or in print, you'd better be able to prove your contention (as @ssaki points out). Either way, don't expect any useful dialogue to ensue after that point.

The noun ‘liar’ is not itself an offensive word in the same way that, say, shit or bitch are — in those cases, the words themselves are (to many people) offensive independently of their meanings. (This is quite silly, as George Carlin probably said best, but it is nonetheless the case.)

With liar, the word is no more offensive or derogatory than its meaning. However, to say that someone is a liar is a real slur on their character: this is the sense in which liar is offensive.

As a native speaker of American English, I would consider it offensive to be called a liar whether it was true or not. You also need to be careful when accusing someone of lying that you don't get into a situation where you are guilty of slander or libel. Even though you consider someone a "proven" liar, the courts may disagree. Laws in your area may differ.

The best way to handling this situation is with diplomacy. For instance, saying,

"I'm not sure that's true."

or

"What are the facts behind that claim?"

, etc. This way you can put the speaker on the spot to back up a claim without actually calling them a liar. In addition, you may get them to admit what they are saying is not entirely true.