Is it possible to run a dual power supply server with one PS plugged into 110V, and the other power supply plugged into 208V? We've asked the vendors and they say they've never tested it or some other "i dunno" type of answer. The power supplies both work at 110 and 208 without any issue, it's the 'mixed' operation that is in question.
Thats up to the server vendor. If you don't even tell us, which servertype you use, nobody could you answer this.
It should work, as the supplies will output the prober voltage anyway. But maybe the server's monitoring chipset don't likes that.
EDIT As you really need the mixed-operation you should test it with an old/broken system first. If there is no trouble, you have a great chance the other systems will stay alive. Keep the mixed-operation as short as possible, but long enough to allow the supplies to make a failover.
Obviously, I have to disclaim any liability for if something happens.
That being said, running them on two different voltages is no different than running them on two different 120v circuits. The power supplies transform the power to uniform outputs, so the only issues that you could run across are things like differing ground references, power factor differences, etc.--the same types of problems that can come up with from different circuits of the same voltage.
Although this is possible, it's not necessarily helpful; with 208, you are using two hot legs of a three-phase, 120v system. In other words, you are doubling your risk exposure versus using only one: if either hot leg fails on the 208, you're down. If your building also had a 277/480 service, that would provide a higher level of reliability, since they are different services (different transformer, feeders, and gear/switching).