I have a few colleagues who do that sort of thing, and the same sequence of events tends to happen over and over again. Someone (generally working from another time zone) sees that employee X is online. They send X an IM asking for help with something urgent. After a few minutes with no reply, they send employee X their request via email. A few hours later, still no reply even though they show as online and available. That person ends up angry that they're apparently being ignored, frequently complaining to X's manager about it. In reality, X is asleep because it's 2AM local time and they're merely faking their online status. In a more extreme case, one colleague got assigned to a midnight conference call because the European sales manager "saw that he was always available at that time of day" and assumed he worked the late shift.
So yes, showing "online" all the time might make you appear like you're working, but at the same time you're giving a false impression. It's a very thin facade that might make you look slightly better, but as soon as someone tries to act on that information it quickly becomes worse than if you were just honest to begin with. IMO, the risk far outweighs any potential benefits.
That being said, don't assume that employees with "green" status at odd times are necessarily trying to game the system. I've used IM systems where someone was always listed as "green" whenever they were signed in on their mobile phone since they were technically able to receive messages. It didn't imply anything about the status of the person, only the status of the software.