On 'modern' Filofax organisers the cover over the mechanism is riveted on, so it isn't possible to see how it works, but on my old Winchester the cover just clips on. Being a curious engineering type of person, I couldn't resist having a look under the cover!
Sorry for the quality of the photos, but taking close-up pictures of 'shiny' things isn't easy!
So here are the rings closed, with the tabs 'up' so when you press down on the tab it presses the two strips of metal you can see running the length of the mechanism.
These two strips you might notice are now hinged up slightly, so the rings open. The strips interlock at intervals along their length on alternative sides (the small half moons on the upper surface). These two strips are contained in a springy U channel, which is allowed to flex a very small amount for the rings to go from closed to open. But it is this channel that is the 'spring' and therefore it is this that keeps the rings closed.
So to ensure you don't damage the mechanism, always use the tabs to open rings and use gentle pressure on both sets of rings to close them.
Now I have seen inside I think I can see what could go wrong with the mechanism with the strips becoming out of their interlock condition and so they would just fall open or at worse come out completely.