Bebilith have never been tanar'ri, even when the term was first coined in 2nd Edition (in an attempt to move away from the term "demon", which set too many of the religious right against the game). By their very description, they're demon-hunters. That's what they do in their native Abyss: They crawl around the various layers of the Abyss and they hunt demons, for no purpose other than it seems to be their natural function. It makes little sense, even amongst the Chaotic Evil tanar'ri, to have a tanar'ri that hunts other tanar'ri as a natural function of its ecology.
However, 3rd Edition has been really good about insuring that types stay pretty consistent, so that a Dungeon Master can assume certain standards when using a creature that's part of a greater overall type like demon or devil or dragon. There are certain qualities that all demons have, and anything under the "Demon" heading in the Monster Manual is assumed to have those qualities. So you could probably assume that, unless its description indicates otherwise, a bebilith is going to have all proper demon qualities. However, there's a type of demon called tanar'ri, and those have their own qualities, because "tanar'ri" is a separate kind of demon. If you'll notice, a bebilith is just listed as a Chaotic Evil Outsider, and it doesn't have the tanar'ri subtype. So no, it isn't going to gain any of the standard tanar'ri abilities.