Once again, they're correct, but unnatural. a. Let the door be opened. The passive construction indicates that the door should be opened by a person. The construction is correct, but it is extremely formal, especially because of the combination of the formal 'Let something be..' with the formal-sounding passive. The sentence sounds like another piece of totally artificial English from an ancient grammar book. In the real world, we'd say something like 'Can you ask someone to open the door?' b. Let the door open. Without the passive form, the implication is that the door will open by itself. 'Let the door open' means 'Don't prevent the door from closing'. If the door keeps swinging open and your friend keeps trying to close it, you could say 'Let the door open'. Are you getting all these sentences from a book, or making them up yourself? Either way, very little of this is natural or useful English.

They are both correct, but I would say that I consider the difference to be what is implied: Let the door BE opened (implies that it is being opened BY someone/something). Let the door open (could easily mean it is being opened by someone, or perhaps opened by itself or just falls open?) I hope this is helpful :-)