Ozone Systems in Pools
TFP does not use or endorse the use of ozone systems. Ozone is not a water sanitizer and ozone systems have little to no value in a residential, outdoor pool.
Ozone manufacturers push Ozone systems by saying you need to maintain a minimal amount of chlorine in the water. Here at TFP we group them together as an "alternative" system that is potentially unsafe to the swimmers in the water.
Additionally, ozone systems are impossible to test to see if they are working or not.
Ozone provides no residual effectiveness. It only cleans what comes in direct contact with it. We would much rather maintain a higher Free Chlorine level and know that pool water is safe since it's the water itself that carries the sanitizing effects along with it.
Ozone Systems in Spas
The biggest plus with a functioning ozonator (not all work properly or put out enough ozone to work well) is that it will oxidize many contaminants in the water, but it's not fast as it takes time to get water circulated through the ozonator. Though spas circulate water faster than pools, it still takes 4.6 turnovers of the water to get 99% of the water through the ozonator and that's assuming no dead spots. You still need a residual sanitizer if you're going to prevent bacterial growth in the bulk pool water.
The biggest minus with an ozonator is that it mostly injects air into the water and that aerates the water. If the ozonator is always on (and many are), this leads to a rise in pH if you use a hypochlorite source of chlorine. So many spa users use Dichlor which is acidic (when accounting for chlorine usage) and this helps maintain the pH, but has the CYA rise.
An ozonator probably makes more sense in a bromine spa than a chlorine spa as it can reactivate the bromide to bromine (though can create bromates -- so don't drink the water) and most bromine systems are net acidic so the pH will be more stable with the ozonator. As was pointed out, it is technically unnecessary if one maintains a residual sanitizer (chlorine, bromine, PHMB/biguanide/Baqua).
Just keep in mind that the sanitizer usage in a spa is MUCH higher than in a pool due to the lower water volume (higher bather load) and higher temperature that causes one to produce more sweat and causes chlorine to outgas more (especially with an ozonator).
If you use your hot tub a lot and keep it covered when not in use then an Ozonator can help, however if you only soak in it on the weekends, etc. then an Ozonator will actually consume more chlorine than the tub would use to oxidize bather waste if you did not have one. Also unlike Ozonators for pools, ones built for hot tubs are relatively cheap. 
Does Ozone Increase pH or TA?
Bubbles from an ozonator are air plus added ozone, so it contains mostly nitrogen gas and oxygen gas with some ozone gas (some of the oxygen has been converted to ozone via corona discharge or UV depending on the system). The amount of CO2 will be that in air and will be less than that in pool water so there will be some degree of mixing and aeration as with any air bubbles going through the water. The ozone will mostly dissolve into the water as an aqueous (dissolved) gas.
The pH will rise more than it would if you had no ozonator, but it's not because of the ozone but rather because of the air bubble that not only pulls some carbon dioxide from the water, but more importantly physically disturbs the surface of the water where the bubbles escape. Such disturbance increases the rate of carbon dioxide outgassing.
The same principles hold for saltwater chlorine generators and their hydrogen gas bubbles except they have no carbon dioxide in them initially so have a somewhat greater effect on aeration.
High-end commercial ozonators have oxygen concentrators feeding nearly 100% oxygen gas to the ozonator to produce higher levels of ozone (6%+) and in this case the rest of the gas is oxygen.