Will a Window Air conditioner rated at 230 volts operate at 208 volts

NEMA standards generally ask that motors operate properly at plus or minus 10% of the nameplate voltage. So, if you REALLY have 208 volts available, you're still within spec.

My old highschool had window shakers rated at 230V and the compressors didnt last long on 208 V probably because the line voltage was slightly below. They ended up adding buck boost transformers with the new units and that corrected the problem.
I have seen (and installed) 230V units on a 208V system that had no problem, in fact they were rated "208-230V" and not just "230V".. They have been fine as far as I know.

A.D

It should work ok. If you have any trouble with the compressor not starting up, find the run cap and add a "hard start" capacitor/relay to it. It will give the initial boost needed to turn the compressor over.

a problem with motor loads if refrigeration equipment is the start up. At 208, it can have difficlt time. The actual voltage from the POCO can make the difference. I had a local businessman running a 230 v freezer on a 208 service. the problem was they were so far from ther sub station, the actual voltage pending on the time of the day could low as 206. At that level it sounded horrible when it started, Athough the voltage was right at the 10% range, it is hard to spin a loaded rotor.

Remember, there's a difference between actual voltage and nominal voltage. US utilities seem to keep this difference very small, but it exists, and if the 208 are rather on the low side you're in trouble - remember, worst case it could be 208 -10%, i.e. 187V.

Many motor manufacturers are name plating for either 240/120 or 240/208/120. On 208 systems make sure the motors are rated for 208. You can see from the issues presented above it can be a problem. I have seen warranty failures denied due to installing 240 volt motors on 208. Check any motor catalog and you will see that the motor industry has addressed the issue.