Specifically, I'm looking at Old Spice and Axe sprays. The labels warn that users should not inhale the fragrance (ironic given its purpose), but what are the consequences of prolonged inhalation?
I can't speak for Old Spice, but Lynx/Axe is basically just a solution of perfume in ethanol. The ethanol kills the skin bacteria that generate the malodour (though the population quickly recovers!) and the perfume just makes you smell nice.
Pure/concentrated ethanol will destroy cell membranes, which is how it kills skin bacteria, so it's more dangerous than its widespread use in drinks would suggest (though fortunately diluting ethanol with water greatly reduces the damage to cell membranes). The solvent in Axe is about 95% ethanol and this is concentrated enough to cause damage. There isn't very much ethanol in a typical few seconds of spray, but I suppose if you pointed the can up your nose and inhaled as you sprayed you might cause some real damage. I suspect the intense pain it causes would mean most people will do this only once!
But I would guess the most dangerous ingredient in the deodorant is the perfume. Perfumes are an unholy mixture of terpenes, alcohols, ketones and lots of other stuff, and many of these chemicals are irritating to skin. Handling flasks of pure perfume is done in a fume cupboard with gloves and protective eye wear. Getting any large amount of perfume onto a mucous membrane like the eyes, throat or lungs would cause irritation and pain. But again, the amount of perfume in a few seconds of spray is small and you'd have to work at it to do yourself any serious harm.
I suspect the warnings not to breathe the spray are largely just being cautious and/or a way of being protected from being sued. Though in principle the contents of the spray can cause harm this would take exceptional circumstances, and even then the result is likely to be local irritation rather than any serious damage.
To my considerable surprise I have found a safety data sheet for Axe deodorant online (note that this is a 55KB PDF). The sheet gives the formulation though it omits the perfume concentration. I would guess this is because the perfume concentration is below the level where it has to be disclosed. The section on toxicology states:
SECTION XI. TOXICOLOGY INFORMATION ACUTE EFFECTS EYE CONTACT: May cause redness or irritation. SKIN CONTACT: Overexposure may cause a skin reaction such as redness. Do not use on broken skin. INGESTION: May cause nausea, vomiting and diarrhea. RESPIRATORY: Intentional misuse by deliberately concentrating and inhaling the contents can be fatal or harmful. CHRONIC EFFECTS: None expected CARCINOGEN CLASSIFICATIONS NTP: None IARC: None OSHA: None