How do I rename an open file in Emacs?

Yes, with dired mode, you can:

• C-x d to open dired
• RET to select directory of current file
• C-x C-j (dired-jump to the name of the current file, in Dired)
• R to rename the file (or dired-do-rename).
• q to go back to the (renamed) file buffer

The rename is equivalent to a shell mv, but it will also update any open buffers, and unlike mv it will not change the access and modify times on the file in the filesystem.