How can you cook multiple omelets without burning them?
I have had the same issue when cooking lots of quesadillas, and have found the following combination of techniques to work quite well. 

The obvious answer has been hit upon already, lower the heat. But I think that's missing an important aspect of the issue. What's happening is most likely that you are turning on the heat and then cooking your first omelet before the pan has actually reached temperature. Depending on the type of pan, it can take up to 5 mins for the pan to reach temperature. So, as you continue to cook, the pan is getting closer to temperature. Of course, each time you add more cold egg, the temp decreases again. So your heat is too hot, but it doesn't really become apparent until the last omelet. Rather than just starting at a lower temperature though, start at the same temp you tried and then turn down the temperature as you continue to cook. 

The pan can also heat rapidly between omelets. Take the pan off the heat, swirl your butter or oil through it, add the egg, and then put it back on the heat. If it seems really hot when you add the egg, you can cook it for a second with just the radiant heat of the pan before returning to the burner. This will stop the pan from getting too hot between omelets.