I have a 9 year old Ibanez RG250 with floating bridge. When I use the tremolo my guitar goes slightly out of tune, goes flat if I push down, and sharp if I pull up (I'm talking of course after I stop using the tremolo). I can get it back on tune if I push/pull a bit in the opposite direction. I have a feeling this is normal even in new guitars, but I was wondering if it gets worse in time, and if that's related to the springs.
There are closely guarded methods to setting up a floating vintage tremolo (non Floyd Rose) so that it stays in tune. I've done quite a bit of research on it (hunted down interviews, videos, books, and asked techs) but I've not seen any solutions/explanations blaming the problem on old springs.
Often keeping a floating tremolo in tune has to do with how you setup the guitar. It can involve more obvious methods like lubricating the nut to reduce friction & stretching new strings, to less obvious ones like adjusting the bridge claw to a specific height.
Anyways, here are two great videos showing two different ways to keep a floating trem in tune:
Over time, I've learned to use a combination of all the above tips and I'm happy to say that I've been able to keep my Strat in tune after whammy use. Give some of these methods a try and see what works for you.. good luck!
With two of my old guitars (one with a Floyd Rose about 20 years old and one with a Kahler about 23 years old) they will go out of tune a bit the first few days I use them with a new set of strings, but after they settle down they'll then be in for months - and I do divebomb and pull up a fair bit:-)
They both have locking nuts, which definitely helps by reducing the amount of string slippage through the nut, and there isn't much behind the bridge either, about 3 mm before the string clamp blocks.
If you don't have a locking nut, you could try replpacing your nut with a graphite nut, or using lubricant.
Over time the knife edge of the trem unit where it rests against the pivot posts can wear down. This can mean that the trem is more likely to stick in certain positions just slightly out of tune. Floyd Rose patents cover a bunch of different things, one of which being the construction of the knife edge, and when they licence the design out this is one of the things that they keep back so that original Floyds have a leg up over copies. This may play a part in your problem.