Yes: If you have flat feet, and are running in shoes not designed for flat feet, you can get sore knees or other orthopedic problems.
If you have flat feet, chances are you are an overpronator, and you should be wearing "motion control" shoes.
I suggest you go to a store specializing in serving runners, and discuss your injuries. Bring your current shoes with you, since the wear patterns can provide some information.
I also suggest you check your library for "Lore Of Running", by Tim Noakes. There are sections on injury diagnosis, and injury prevention.