In short, yes it does oxidize, but it's not usually a problem for your beer, since it's heavily diluted.
With no airlock (to allow gas transfer) and constant stirring, oxidization is inevitable once the yeast have come out of the lag phase and start fermenting. During the lag phase, most of the oxygen is taken up by the yeast as part of propagation, but after then, the yeast prefer to produce energy anerobically and they start to ferment the wort. Fermentation (Wikipedia) says,
even in the presence of abundant oxygen, yeast cells
greatly prefer fermentation to oxidative phosphorylation, as long as
sugars are readily available for consumption (a phenomenon known as
the Crabtree effect)
So, once the fermentation starts, there's little additional oxygen takeup by the yeast so long as there are sugars around. The constant agitation will speed up the oxidization of the wort, and so some oxidization will have occurred before the yeast have consumed all the sugars and switch back to aerobic energy production.
But, is the oxidization a problem for the final beer? I'd guess not. If you pour off the starter wort (as is good practice), then what is left in the slurry is probably diluted about 50-100 times when pitched. With that amount of dilution, off flavours are going to be much less pronounced, but of course it depends upon how oxidized it is and how much flavour will be in the final beer. If you wanted to play it safe, you could make up a dilution the starter wort to 1/50 strength and see how it tastes, but I'm guessing washing will not be needed.