I suppose if the one or two defining features showing you're a Genasi that you chose is that your hair is fire or something then maybe be very cautious? But most Fire Genasi from their race blurb aren't really on fire like SladeTracey said. Mostly it's ashy skin colour or a smell of burning or something superficial like that.
The only reason my Fire Genasi doesn't drink is because she's 11 and no-one will let her.
Even if they were hot enough to light almost pure alcohol on fire that doesn’t mean they’d be hot enough to light a drink on fire. I don’t know if you can light normal vodka on fire or not, but I know that if you can it’s not that easy. Barcardi 151 on the other hand has a warning on it that if you’re not careful you will light it on fire. Even an Efreeti isn’t hot to the touch unless they want to be and a Fire Genasi has one Efretti as an ancestor.
I’d still definitely let a Fire Genasi PC do the circus fire breather routine with alcohol without needing to use a torch to light the alcohol on fire. It’s great for fun and it won’t unbalance anything.
You can light a lot of alcoholic drinks on fire without a wick, like sambuca for example. With the sugar in the sambuca sort of acting like a wick I would guess. If you have it in your mouth you can even light it on fire there. But when you swallow it or close your mouth, the fire goes out.
Also, I am amazed you believed you would find any information about that. You have some imagination!
Wine, beer, and fair number of distilled liquors do not have a high enough alcohol content to burn, so the gensai could drink them without issue. Higher proof alcohol might ignite, but would only burn as long as there was oxygen available. So closing his mouth or swallowing would put the flames out.