The UAE has granted amnesty to its residents who have overstayed. Under the amnesty, you could either pay all fines and legalize your status or request an outpass and exit the country without paying the fines. I can't post the link to the gulfnews article, but its a two month amnesty from Dec to Feb...
Doha is pretty close to Dubai may be they were afraid that you will come back somehow by land. I don't know... That's exactly what my PRO said Did you go to the Immigration Office? How big is the fine? Well I was supposed to exit no later than 2nd of Jan. so it adds up to 2,400dhs I haven't been to...