First of all, if you are American you don't need a "visa", you need a "permit" to study or work in Canada. A visa is an entry document. A permit says what you can do when you get here. American are exempt from requiring a visa to enter Canada. So if you flew here for vacation you don't have to apply for a visa, simply travel on your American passport to enter for a visit.
If your application for permanent residence was successful due to the sponsorship by your boyfriend, no, you would not have to return to the US after your study permit expired. Permanent residence voids all temporary permits so your study permit would become null and void.