Syntax errors coming from your web application mean there is a page somewhere with a bugged feature, or perhaps the whole page is broken. Of course that's worth following up on?

Edit: maybe I should add a concrete example. I semi-regularly look at the apache error logs for some of my hobby projects (mainly I check when I'm working on it anyway and notice another preexisting bug). I've found broken pages based on that and either fixed them or at least silenced the issue if it was an outdated script or page anyway. Professionals might handle this more professionally, or less because it's about money and not just making good software, idk