In the UAE and Dubai, it is against the law to live together, in the western sense, with someone you are not married to. You can only live together with a member of the opposite sex if you are married to them, or they are a family member. So the western expression about "living in sin" is taken more literally in the UAE.
Dubai police do not spend their time walking through apartment complexes and hotels knocking on doors and asking for marriage certificates (although in Sharjah they do according to news reports in April 2010), and there are no morality police as is the case with the Mutawwa in Saudi Arabia. However, if someone makes a complaint about illegal activity, then the police are obliged to investigate, and if the complainer has more wasta than the police and/or the alleged criminals, then the police will naturally investigate the complaint even more thoroughly.
Strictly speaking, it is illegal to be in a private room, or even in a car, with an unrelated member of the opposite sex. This applies to rooms in Dubai hotels also. The rule is referred to as the Tawajed clause.
In reality, the situation is not quite so harsh. This is what one official was reported as saying in November 2007 (in a Gulf News article - see Unmarried couples living together in Dubai forum topic for more comments on Sharia law, culture, customs etc with respect to men and women mingling in the UAE): "Mohammad Yousuf, Deputy Director of the Dubai Courts Department, said
while the Federal punitive law is implemented in the UAE, the situation is different in Dubai. The Tawajed or Tahseen Al Ma'asiya clause, he said, is not implemented in Dubai."
It is almost unheard of for anyone to get into trouble because they're sharing a car with someone of the opposite sex that's not related to them. If there is a problem, it's more likely because of another reason, for example the police think an illegal taxi service is going on, or there's been an accident involving alcohol.
Hotel rooms are often shared by unmarried couples. We've never heard of a hotel asking for a marriage certificate, and they usually only want to see one passport.
Even if both passports are shown, eyebrows wouldn't be raised at the different surnames since it is usual in the UAE for Arab women to keep their own surnames after getting married.
http://www.dubaifaqs.com/living-together-in-dubai.php