Well, yes, burning eyes can be a sign of diabetes. Generally speaking, it can lead to floaters in your eyes and that will make you feel annoying. Also, when having it, you will suffer from weaker blood vessels. And then that may lead to some high potential for problems later, such as blurred vision and floating spots, even burning eye. So just pay more attention to it.
Well, I can see that you are quite concerned about your health problems. So, from what I know, it is true that sometimes diabetes would give rise to some problems with our eyes when it becomes serious. The symptoms include painful eyes, swelling eyes or burning eyes. Burning eyes is not necessarily one of diabetes' symptoms, but it could be generated by diabetes. Hope you recover.
Maybe, diabetes can cause the burning eyes. Although the burning eyes are the symptom of diabetes, they can be caused by other reasons. If you work in front of the computer for a long time, you could get the red and dry eyes which will make your eyes feel like burning. You should take care of the eyes and use some eye drops to release the symptom.