Not only is this quite normal for a singer, it can actually be a sign that you're breathing correctly in the more intense passages. I first experienced it when learning to breathe from the diaphragm when I first began taking voice lessons. When we sing intensely, we tend to go through a lot of air. Compared to normal breathing, this looks an awful lot like hyperventilating... which is precisely what it is.
The effect you're describing is called respiratory alkalosis. It's a normal response to hyperventilation: you're expelling carbon dioxide from your body more quickly than it's being produced. This overwhelms your blood's pH buffering system and leads your blood to be slightly alkaline compared to normal (carbon dioxide in solution is quite acidic, which is why Coca-Cola and other fizzy drinks are so acidic).
This effect is not at all dangerous for an otherwise healthy person, although it can be unpleasant and even scary. You do stand a risk of fainting if you overdo it, but you will faint long before you cause any kind of real damage. (I'm not a doctor, but I confirmed this information with one.)
This should improve somewhat with practice, both due to tolerance to the effect and due to learning to be more economical with your breathing. If it continues to bother you, do some exercises specifically designed to conserve air while singing (one such exercise is to deliberately sing a passage for as long as possible without taking another breath -- start with 2 or 4 bars and work your way up).
All this said, if you experience frequent dizzy or fainting spells when breathing normally, or if this doesn't improve at all with practice, it's always better to see your doctor and get a medical opinion specific to you.
The previously-mentioned doctor had this to add: "The only other thing is that if he's trying to force out the note [i.e. singing with some restriction in the throat], he's doing a modified Valsalva, which stimulates the vagus nerve. That drops blood pressure, causes nausea and lightheadedness as well."
Singing with restriction in your throat, which is a common, but incorrect way of getting a more intense sound can also cause this. Still not dangerous, but potentially a sign of incorrect technique in some contexts.