The only mosque in Dubai that non-Muslims can enter (at specific times) is the Jumeirah Mosque. Tours are run throughout the week hosted by the SMCCU (Sheikh Mohammed Centre for Cultural Understanding). Visitors are shown the ritual of wudu, a cleaning before entering the mosque, then taken through a sort of Islam 101 lecture before being encouraged to ask questions at the end.
If you are female and are wearing clothing that is inappropriate (bare shoulders, skirts above the knees, etc.), there are abayas (full length dresses for women) for you to borrow. Although not required when I was there, it is suitable to bring a scarf you can cover your hair with.
Fun fact: Our guide told us that there is a mosque every 500 metres in Dubai, accounting for approximately 1,600 mosques in the city.
*My trip was hosted by Emirates and included three other Canadian travel writers.