I was skeptical myself about the existence of the danger triangle; I just asked my father, who is a physician, today, and he told me that this indeed is true, describing an area that covers the eyes (and brows), the nose and the upper lip. (Actually more like a square). He further told me that it leads to cerebral sinus venous trombosis.
If you want to know what this cavernous sinus is, Wikipedia is detailed on that (and does mention the facial area, too).
Cavernous sinus thrombosis (CST) is the formation of a blood clot within the cavernous sinus, a cavity at the base of the brain which drains deoxygenated blood from the brain back to the heart. The cause is usually from a spreading infection in the nose, sinuses, ears, or teeth. Staphylococcus aureus and Streptococcus are often the associated bacteria.
Anyway I'd still count this is just as anecdotal evidence and he never had a case of this to happen.
But with the help of the correct term I was able to dig up some results and there actually seem to be some cases, first were described even around 1852 (I assume that the facial cosmetics industry wasn't as powerful back then, but what do I know?)
Infections of the "dangerous area" of the face were first mentioned in the literature in 1852, when Ludlow described six cases, three of them fatal. The first patient, a moderately stout man of 20, scratched the head off a pimple on his lip six days before admission to the hospital and died 36 hours after admission, in spite of the use of compresses, sedatives and leeches. […] In the first case autopsy revealed a purulent exudate in the lungs and kidneys […] In neither case was the head examined.
Another paper cites the same work but goes further:
From 1922, eighteen cases of primary infection of the nose, lips and face were admitted to the wards of Lebanon Hospital […] Of the six patient with severe symptoms who dies, four were males and six females, varying from twenty-four to fifty years of age. [...] In all six there was a positive staphylococcus aureaus blood culture, with the subsequent cultures becoming progressively worse, with thrombosis of the cavernous sinus terminating in death.
So I think I would say: Yes, popping pimples in the face can be deadly. Although cases are not very common and while reading across the lines treatment has improved, it might still be dangerous.