Are there any technologies that go missing after the destruction of the library (similar to the loss of Greek fire that happened 600 years later)?

Well, it is hard to say, since direct technologies leave as more of an archaeological record as say math texts which are essentially technologies. But one can't help but notice that the Antikythera mechanism predated the library, and we didn't have anything like it again until the 1400s. However, this is imperfect in that it looks like a lot of wars and problems occurred between the height of the Greeks and the burning of Alexandria so pinning this sort of thing on it is tough.