I have returned from my trip and would like to share some of my insight. The other answers here are similar to what I experienced so I thought it would be useful to share additional information that I wish I had known before traveling. Note that I stayed in the Dubai Marina area which has many more Westerners than older parts of the city (such as Deira) and is less conservative in general.
I would like to say that doing a few mosque tours gave me a much greater appreciation of Ramadan and Muslim culture in general. Ramadan is as much about following religious principles as it is about being aware of our own habits and the things we take for granted. It is a time of giving and family (such as the communal Iftars in the evening). If you view Ramadan in a positive light rather than an inconvenience I think you will have a much better time in the city and find it easier to respect the cultural norms.
Many of the restaurants in the New Dubai/Marina area offer carryout services with special Ramadan hours. Many also offer free delivery. If you plan ahead a little, it is easy to order and bring it back to your room to eat if you don't want room service (or in my case I was in an AirBnB). Just be careful when you are in elevators or crowded spaces - Muslims should not even smell the food you are carrying around.
I read on some of the links above that some restaurants have covered areas that are considered "private" so you can still eat there. The only place I saw them was in the Atantis resort on the palm.
Plan your day carefully (huge breakfast) and bring non-perishable food + water in your bag when you travel. It will be very hot and you will be hungry. Bathrooms are found in every metro and extremely convenient for stopping for a quick (quiet) snack. Just be respectful of those around you.
If you are planning to do any sort of tours they will generally still allow food. I participated in a desert safari and a boat tour in Oman and both still offered food since we were in a private area away from the main city.
In the new areas of Dubai as well as the malls, you can probably get away with quite a lot (but that doesn't mean you should). Many Western females there wore short shorts, spaghetti straps, and other common summer styles. In Deira however (an older part of the city), my girlfriend was dressed very conservatively (long pants, long-sleeves, and all her shoulder/neck hidden) and still got full-body eye-glances from 90% of the men there (the souks there were almost exclusively male-populated as well).
As far as PDA goes, I saw perhaps 1 in 10 couples holding hands in the malls and nothing more. In the malls and newer areas in Dubai you can probably get away with it but it's better to be respectful. For a more extreme case, in the Abu Dhabi Grand Mosque I put my arm around my girlfriend for a picture without thinking about it (completely friendly gesture, like I would do for any male friend for example) and we were told off politely.
Ramadan Time Restrictions
Prayer times probably will not interfere with most of the things you are doing around the city. Many places are closed during the day but there is still plenty to do. Historical places, mosques, and malls are still open. However, many souks will have odd hours or be closed entirely. Note that the Metro might have restricted hours (on Fridays it opened at 1 pm). Do not eat in public until after 6:45 - 7:00 pm.
If you are an unmarried couple it might be a good idea to bring some fake wedding rings along. Many of the resort hotels probably won't ask you anything but you might get some questions if you are outside the newer Dubai areas.