There are a few misconceptions going on in here in regards to a yeast starter, which this thread isn't even about in the first place. But, let me try to clear those up:
1. It is OK and even encouraged to constantly aerate a starter. More oxygen generally means more yeast growth which is one of the main reasons to do a starter. You're not going to drink your starter, so the affect this has on the taste of it is irrelevant.
2. Given #1, you generally do not put an airlock on a starter. You put a piece of loose fitting aluminum foil on it or one of those fancy spongy plugs. That will let the air in but keep airborne bugs out.
Ok. Now for the actual topic - I see no theoretical reason why you couldn't use a stir plate when fermenting actual beer. I'm thinking it really isn't necessary 99% of the time, though. The strength of the fermentation is usually such that beer will be getting swirled around plenty. But, if you want to get a bit higher attenuation or try and prevent a stuck fermentation, it could work. MikeSzwaya does say he uses one if he wants a beer to finish really dry. Given that it can increase attenuation, it may actually prolong fermentation rather than making it finish faster, though.
I see some practical reasons that may make it difficult. For one, you need a stir plate that can withstand the weight of your container and 5 gallons of beer. Second, I think you'd need a pretty big stir bar to sufficiently stir your entire batch. Third, you'd need the power on the stir plate to move that large stir bar fast enough. All these things are irrelevant, though, if you're doing smaller batches.