Well, yes of course, sometimes I have the same problem with you since I slept late and got up quite early in the morning. This is the most common cause, however, sometimes we will have burning eyes in the morning due to some other reasons, red eyes, infected eyes, and some other. Therefore, the best strategy for you is to identify the real cause of your problem, then take some necessary actions against it.
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.
Yes. it is possible to feel burning eyes when you lack of sleep. As you described, you went to bed very late last night, you must feel very eye strain and tired before you go to bed. And also, your not got enough sleep that means your eyes not get good rest. So, you feel burning in the morning. In fact, some people also tears in the morning because of tired and lacking of sleep. Nevertheless, eye burning can also caused by infection and other eye problems. SO, if you can't recover after a good rest. You'd better get your eyes checked by a doctor.
Yes, it is caused by the fact that you went to sleep too late. It may be all about the lacrimal gland, which is related to sleep. I think it's the same mechanism that happens when you're stay up late at night and your eyes feel tired then. The lacrimal gland stops flowing out tears if you lack sleep enough.