Which mosques can be visited by non-Muslims in Abu Dhabi and Dubai?
Are foreigners and non-Muslims restricted to certain times? What about certain dates?
Are the different restrictions for visiting the exterior, grounds and interior?
According to the Dubai authorities, the Jumeirah Mosque is "the only mosque in Dubai which is open to the public and dedicated to receiving non-Muslim guests." (At least insofar as going inside is concerned, I can't speak to the grounds, and you can certainly see mosques from the outside around the city.)
They operate a tour most days:
The Sheikh Mohammed Centre for Cultural Understanding hosts visits of the Grand Jumeirah Mosque 6 days a week Saturday through Thursday at 10:00am. You do not need a reservation to attend this visit, but you must arrive at the main entrance to the Jumeirah Mosque by approximately 9.45am. Each tour lasts approximately 75 minutes. The tour cost 20 AED per person including water, dates, Arabic coffee, tea, and traditional pastries. Children under 12 are free. Parties of 10 or more should contact the SMCCU prior to your visit.
Modest dress is preferred, however traditional attire can be borrowed from the Mosque.
If you want even more, the page goes on to indicate that it is possible to arrange a private visit of the Jumeirah Mosque, along with several other mosques, if you contact them in advance.