How do I know when to replace a violin E string?

Sometimes if a string looks fine, it may be bad. Check the harmonics for clarity, also check the 5ths.

If an E string has discolored or the gold plating has warn off it is a good idea to replace it. I find that E strings generally need to be replaced sooner than the other strings. Packaged E strings do not have a shelf life unless they are several years old. I have seen old packaged E strings in old cases, that have rusted. Needless to say I would not want to use one, or at the very least get a tetanus shot first. Ha Ha

Of course, gut strings, even wound gut strings have a shelf life. Since they are an animal product they start to break down and deteriorate over time.