a Dubai and Qatar Residency Permit, you are illegally in either or both countries. You are not allowed to hold permits for 2 GCC countries at the same time.

You will still need an exit permit to go out of Qatar.

I don't think you need an exit permit to leave Dubai.

To answer your previous question yes, you can use your Dubai RP to get into Dubai and your Qatar RP to get into Qatar - even if its illegal.