It's probably easiest to reinstall iTunes.
If the firewall does not remember your choice, then the digital signature (Code Signing) of iTunes is broken. By default, the OS X firewall accepts any software that was signed by Apple without even asking you. So: it should not have asked you. And once asked, it should be able to remember, based on the digital signature. A broken signature (for example caused by changing the contents of the iTunes.app package) will prevent all this.
(Did you by any chance manually change
the iTunes icon the application somehow? Or did you install some plugins?)
You can validate the signature of your iTunes by copying the first line and paste it in Terminal:
codesign -vvv /Applications/iTunes.app/
/Applications/iTunes.app/: valid on disk
/Applications/iTunes.app/: satisfies its Designated Requirement
(More details at If Mac code signing is tampered with, what might fail?)