It all depends on how much RUST your talking about, in most cases NO, We use 3/in/1 oil with 4/0000 steel wool in the rusted area, so not to remove the bluing give a good wipe down with Rem-oil or what ever oil you use.
3in1 oil has K1/Kero in it,it lifts rust off steel.

Most definitely. Rust on the barrel near the muzzle can cause friction against the bushing and interfere with the lockup.
The shooter may have to push the slide forward with his free hand to lock the action.
That area must be clean, highly polished and well lubricated for reliability.

This question reminds reminds me of a rust related story. As we all know humidity in the tropics causes rust quickly if one is not vigilant. This situation is magnified by local culture. Manioc (Cassava) root is a basic source of carbohydrates in developing countries. It is highly toxic and must be cooked and pounded before being mixed into most daily meals. This toxicity plus use and handling of huge amounts of salt help create hand prints of rust on a rifle barrel if it is handled by a local, but well meaning person in your tropical camp. Another reason to wipe down your firearm at night.
The exteriors of my oldest 375 and 416 show a lot of use, but the actions, crowns and interior barrels are spic and span.