I'm thinking of taking the online/offline icons off my forum and replacing them with text. Will the constant changing of online to offline status of my users be good for SEO?
No I doubt this would help seo. Search engines do like frequently added new content and content updates, but they may be smart enough to see that this is not new content or actual updates. One downside of doing this is that now each thread will now have the words online & offline repeated many times. The high keyword density for those 2 words may cause problems. It looks spammy to search engines and it may dillute the theme/focus of your forum.
Every little bit matters to beat the next guy, when you get to the higher end of SEO optimizing your site, you begin to look for smaller and smaller details.