My director was wondering if changing batteries (such as switching from regular double A to rechargeable batteries) in a microphone could possibly affect the sound quality. I don't think this would be an issue, but if it could be, or if there is an explanation for why this is a possibility, please share.
After extensive attempts, we've bailed on rechargeables entirely and have decided that fresh alkaline batteries are simply a cost of doing business. In digital units you can have the problems Iain McGregor mentioned above. With analog units, you can add reduced headroom. Our biggest problem was simply unpredictability of service life.
The problem lies with the discharge patterns of these two types of batteries. Alkalines maintain something near their rated output for most of their service life and then sink quickly at the end. You can learn how long to expect an alkaline to last pretty easily. Rechargeable batteries, however, sag from their rated output steadily as they discharge. The rate of sag increases as the batteries age, sometimes quite unpredictably. You usually figure out the hard way when your expectations are no longer valid by having them fail in the middle of a scene or show.
If batteries are not providing the full voltage then you can sometimes reduce the range of a radio mic belt pack or make it more prone to interference.
I've had the batter going low on my phantom power unit greatly affect the sound quality. Really, its the less than full voltage scenario you have to watch out for.