Well, yes, generally speaking, lack of sleep can lead to burning eyes. As we know that when you do not have a good sleep, it can disorder your system, also have some effect on your eyes. For example, it can lead to eye strain. Or in some degree, it can also be serious, when you have burning eyes. Anyway, you should pay more attention to it. To treat it, you can use eye drops, which are a very good idea. Also, you can use a thorough rinsing with cold water, so that it will not irritate the eyes. I would suggest that you should not sleep with the contacts, because it will damage your eyes and make the eyes infected. And also, cleaning your contact lenses carefully is also important.