It depends on the rechargable battery. When they run out of power, they tend to do it suddenly, rather than fading out more gradually, which will only be a problem if you start a gig with a low battery.
But another factor is more design-dependent. Rechargables typically run 1.2 volts, while others are rated at 1.5. The "9 volt" ones also run a bit low, I'm thinking around 7.8 volts. In the highly unlikely event that some total bonehead designed a preamp or equalization circuit for the pickup which expected to have more voltage available than the battery supplied, it would be possible to run out of headroom and end up with your wave being squared somewhat. But I doubt that you'll ever find a pickup like that in practice, so don't worry about it unless your guitar sounds like you're playing through a really awful fuzz every time you play especially loudly.
+1 on active pickups (or any other guitar electronics which require a battery) being a PITA which is better avoided if a high-quality passive will do the job.
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