There is a note on the door and on it is written that

It's fine. Grammatically there is an ambiguity, but once you've mentioned a note, anyone not being deliberately perverse will understand that the message is written on the note and not on the door.

Even though the note can not speak, we often write, there is a note on the door which says...

Or, the note on the door indicates that...

There is a note on the door saying "..."

In the quotation marks, you could report the exact words the note says.