Can exercising with additional oxygen burn more calories than without extra oxygen?

No: Unless you have a lung or cardiac problem, you get all the oxygen required to metabolize calories. The addition of oxygen beyond what is required for the oxidative metabolic processes involved in exercise will not "burn" more calories. Extra O2 will cause a house fire to burn hotter, but doesn't work that way with metabolism.

Not likely: In an otherwise healthy person, your blood is pretty well saturated with oxygen at room air and so using additional oxygen would have little effect at all. Some people will try to train at high altitudes with low oxygen as a way to increase their hemeglobin and oxygen delivery to tissues and there is some thought it can be helpful in endurance training.