it depends upon the dimmers. there are both 110-120ish dimmers and 208 - 220ish dimmers in the 2.4 kw size. you should ask the rental company. the generator will probaly need 3 legs plus neutral n ground. Make sure it can handle the amps.
You say 208V, but what is that referenced to? Remember that voltage is not an absolute value, it is always compared to something. You also don't mention what kind of power the dimmer pack requires. For instance, the dimmer packs I used in high school required three phase/five wire 120/208V power.
As far as that "208V" generator - my guess is that the 208V you specify is between hots - that is, the generator generates three phases of AC, each of which measure a voltage of 120V to ground and 208V from hot to hot. Chances are, it will work fine for you, as long as it provides outlets that are 120V to ground, as well as 208V hot-to-hot.
If there is any doubt, ask the rental house to tell you exactly what kind of power the dimmer pack requires, and what kind of power the generator provides. If they don't match, tell them to get you something that does match.
OK, the reason they told you it was a 208 dimmer rack is beacuse its 3 phase . . which means that your rack needs 3 hots, 1 ground and one neutral to opporate properly. Once hooked up PROPERLY, you should have 120* going through your dimmers. Which means that you can use 120 fixtures on it. The only time that you should ever use 208/220 fixtures is when you are wither outside of this country . . . using equipment manufactured for outside of this country, or specialised equipment designed for 208/220. A great example is intelligent lighting, most fixtures can be switched from 120 to 208 with a jumper inside.
Smaller dimmer racks often have the option of switcing between single phase three wire (120/240v) and Three Phase (120/208v) power supplies. Dependant upon what type and size of generator you have available you can also at times have the rental company adjust the dimmer pack for what type of power you have.
Think it's all kind of a moot point, the generator will be what it is and he will have to switch to what ever power source will feed it. Three phasepower supplies would only require 160 amps per leg verses a two pase (single phase) would require 240 amps per leg.
What generator you rent should be sized for what you need - in this case a 24 way 2.4Kw dimmer pack. The three phase output would be more efficient but otherwise won't make a difference once powering up the lights.