The invitation trope is folkloric, but TLB's twist on it remains unique AFAIK -- i.e. that they can come in whenever they want, but the invitation "renders you powerless." I still think it was a cheat to take it as far as Max passing all the tests at dinner, since nobody else has EVER interpreted it that way, so it amounted to a big "ha-ha" on the audience at the end when there was no possible way to think his being a vampire was still on the table.

Someone beat me to the cinematic venerableness of bursting into flames. Pretty sure it is cinematic in origin. I don't recall it featuring in any pre-cinema fiction (the literary Dracula could actually move about during the day, albeit weakened), and certainly not in folklore.

The BtVS "vamp out" look definitely harks back to these guys! And was much more effective there, with the whole demon thing, than when it was used in the Rick Springfield Nick Knight pilot, where it just looked silly. But I think we have Hammer to thank for the "vamping out" trope itself -- it's just usually limited to eyes and fangs, rather than the whole face changing. I don't recall the latter prior to TLB, unless you count transitional shapechanging forms like in Fright Night.

I'm pretty sure both Spike and David owe their style sense to Billy Idol. :-)