Straight from "The EXUP Brotherhood"
The EXUP valve can seize up after a few years (although mine hasn't yet on my '93 bike). More serious a problem is potential failure of the bearing behind the gearbox sprocket, partly caused by running a worn or over-tight drive chain (again no probs with mine yet!). Requires a full engine strip however to sort out!
Rear wheel bearings can be short-lived but are easily changed. Pitted head races tend to be common on older models, causing vague handling and reduced stability, but changing them is easy enough too.
Calliper corrosion affects all pre-'94 models and Yamaha UK even produce a leaflet about this. I have replaced ALL the brake callipers on mine with Harrison Billet units, but not everyone I know has changed theirs. Front discs also have a tendency to warp (which mine did and were immediately replaced with a pair of Brembo's). Worth changing to braided steel brake and clutch lines whilst you're are it (but be sure to use s/s nuts NOT anodized alloy ones which are prone to cracking).
Regular oil changes are a must, especially if the EXUP is used a lot in town as clutch wear contaminates the oil and can lead to blocked oilways. UK bikes were restricted to 125bhp, but the inlet rubbers can easily be trimmed to obtain a few extra horses (there must be hardly any EXUPs that AREN'T de-restricted!).
Also fork seals have a habit of giving up the ghost (although again I'm still waiting for this to happen on mine). You'll probably find that the general quality and standard of nuts and bolts on the FZR1000 isn't terribly good (a case of Yamaha spoiling the ship for a ha'porth of tar), so it's well worth replacing some of these for stainless steel or aluminium items.
The standard clutch basket is not particularly strong, and won't stand up well to constant wheelies or drag starts. Often this is replaced by a billet clutch basket, or the original unit can be modified for better performance.
Despite what I have said above, all-in-all you should find the EXUP to be a pretty damn fine, reliable motorcycle, which looks great, handles beautifully and goes like stink! You should also be able to pick up a very good one for a very reasonable price too, which is another reason to consider getting one. However the best reason on earth is that you will then belong to the world-wide EXUP Brotherhood, so what more do you want?!