This issue could only come up in a forum like this that has members from all over the US and beyond. After all, if people refer to "BPL" in their daily lives, they are referring to their own local library and everyone in their city or county (or whatever entity the library is named for) knows it.
On a forum like this, as long as it's clear from the context which library is being discussed, I don't really care if the poster is discussing Boston or Boondocks
Libraries aren't the only institutions with such abbreviations. There are also multiple universities with the same abbreviation, such as ASU or MSU. How would you determine which ASU, MSU, etc. is the only correct one? And do you think you could really get students, school officials, alumni, etc. associated with all the other ASU's or MSU's to stop using those letters?
http://acronyms.thefreedictionary.com/asu
http://acronyms.thefreedictionary.com/MSU