If I am a US citizen in Tel Aviv, would I be able to sit in the Shaar Zion library and do work? The library website says, "Tel Aviv-Yafo residents can utilize the city’s libraries without charge upon presentation of ID card and resident card". What about non-residents—are they allowed in with some sort of registration, or would I not be allowed in at all?
You can enter the building if that's what you're asking. The WiFi is free, but other services require subscription. Here's the price list (ILS). For books only it's 350ILS (for Shaar Zion, it is more expensive than others), for books and digital media it's 400ILS (all the same).
To register and set up a subscription you'll need to get there and do it in person (residents don't need to pay and can register online).
tl;dr: "without charge" regards book loans only.
Tel Aviv-Yafo residents can utilize the city’s libraries without charge upon presentation of ID card and resident card
means residents can check out books from city libraries without charge. Libraries in Israel generally do not check for residence or citizenship status for visiting and working in the library (if they have a work area, that is). In fact, it is culturally inconceivable to restrict library access by place-of-residence - so much so that the municipality did not consider your interpretation of their notice and thus failed to clarify the wording.