You can use both close and shut with doors and windows:
▶️ Please close/shut the door.
▶️ I closed/shut the window because it was getting cold.
(the past tense of “shut” is also “shut”)
With eyes and mouths, “close” is probably a little more common than “shut”
(especially with mouth):
▶️ He closed his eyes and tried to sleep.
▶️ The dentist asked me to close my mouth.
“Shut your mouth!” is a very rude way to tell somebody to stop talking.
When talking about a store, bank, post office, etc., use only close:
▶️ The bank closes at 4 PM.
▶️ What time does the post office close?
▶️ Attention shoppers: the store will be closing in 15 minutes.