When there is a conflict between how you would expect an abbreviation to be pronounced judging from its spelling alone, and how you would expect judging from the sounds of the larger word, then it can go either way. Or in another direction entirely.
If we considered such abbreviations to be words in their own right then we wouldn't be surprised at this. Related words aren't always pronounced in matching ways, after all. Some also are pronounced in matching ways in some dialects and accents, and not in others.
The only real difference, is that if we are still thinking of it as an abbreviation (unlike say laser which most of us think of as fully a word rather than an initialism or amphetamine which most of us think of as fully a word rather than as an abbreviation for alphamethyl-phenethylamine), then they are probably quite recent.
Further, abbreviations can be coined in text quite separate from speech (people will still write etc and yet pronounce it et cetera and that's a very old one), or coined in speech quite separate from text (people form shortened forms of anything they regularly talk about, but often would still use the full form in writing). This in itself will have an effect on whether seeing lib will make you think /laɪˈb/ or /lɪb/.
There's also in this case the fact that we have several other abbreviations spelled lib. We've the long-standing ad lib for ad libitum, lib for Liberal (the name or part of the name of several political parties around the world), lib for liberation (as in "Woman's Lib", "Gay Lib" though that term has fallen out of favour, make of the lack of interest in liberation, what you will).
Of those matching the pronunciation of library, we've only a half-point, since Libra is sometimes pronounced /ˈlaɪbrə/ and sometimes /ˈlibrə/. (I've personally only encountered that as text-only abbreviations, pronounced as if they was written in full, and anyway).
So with so many cases of lib already pronounced /lɪb/, the odds where in the favour of that becoming the pronunciation used.
At least as far as I've heard. I've also found that abbreviations are more likely than other words to produce not only claiming that they pronounce it differently, but insisting quite strongly that they've only ever heard it that way in their lives.