It will start feeding back into the training set, corrupting things. OpenAI will have an advantage at first as they can trivially filter out everything they have generated from the future training corpuses, since you can only run it through their servers. If they or someone else has breakaway progress such that almost all generated content is from their own servers because users only use them because their results are so much better, they could form a strong self-reinforcing moat against competitors forced to train on their semi-spam which they can trivially filter out.
It's also possible we'll see something like the existing big-tech patent cross-licensing agreements, where they all agree to share their generated outputs to filter from training, making it very hard for new entrants.
Other companies will begin having advantages as well, depending on how well they can get less tainted user data. Think of Discord, for example, where users may use AI but are less likely to gamify it like stack overflow and flood it for points, and instead be correcting its output etc. in programming discussions.
As things become more accepted Microsoft will probably eventually sell access to private github for training, with some stronger measures around avoid rote memorization.