It can if the rust extends above the surface of the chamber to reduce its overall size. I've had good luck removing rust from chambers by soaking with a good solvent and then rotating a large bronze shotgun bore brush (size as appropriate) in the chamber using a drill or similar tool.
Remember to never store your rifle with a snap cap or anything else in the chamber. Anything in the chamber while the rifle is stored is a breeding environment for rust.
Rust in the chamber can cause many problems. Rust in the chamber would have to be heavy to change pressure much and the case would not go in and chamber. But it could show pressure signs that may not be real. In a rusty chamber the roughness will lock the case at its fired location. If the case was shoved a few thousands by the firing pen in firing the case would lock in the chamber, Then the case brass would stretch allowing the base back to the bolt face, Making a possible stretch ring just in front of the extractor grove in the case. Also allowing the shell head a running lick coming back to the bolt face may show more signs of pressure than true pressure. Rust may cause stiff bolt movement on the opening of the action and hard extraction of the case. Cleaning the chamber by some previous suggestions will help smooth the chamber. Good Luck
Once you get it cleaned, try leaving a film of lubricant on the surface of the chamber during storage. You can run a dry patch over the chamber to clean it out before firing it later on. For rust prevention you'll want something with low viscosity. I like Kroil. But a high quality automatic transmission fluid is less expensive and works very well.