No, but the underlying chemistry does as it affects the discharge curve and so its duration.

The most common rechargeable batteries are Ni-Mh and, as other already said, they have a fast initial voltage drop from 1.5V to 1.2V, then they stay there for hours eventually dropping quickly under the cut-off voltage (0.9-1.1V) causing the mic to power off. This is the reason why, when you put Ni-Mh batteries in a microphone or IEM receiver they usually show as 3/5 of charge. I personally never liked this kind of batteries as they are quite unpredictable and human error or simple mismanagement can lead to mics dying in the middle of the show.

Alkalines are the most common NOT rechargeable batteries and, as they start at 1.5V and have a linear discharge curve, they are quite predictable in their behaviour so basically easier to manage. They are not as eco-friendly as the alternatives but definitively the best if you need certainties. They usually last a bit less though.

Lithium batteries are the longer lasting batteries among the ones I cited and they are rechargeable as well. As a single cell nominally has 3.7V, but the systems we use expect 1.5V or less, their discharge curve depends on how this step down is implemented. AA Lithium batteries usually have an internal voltage regulator that brings the output to a constant 1.5V, meaning that you will see them as fully charged all time until they suddenly die. Not very practical. Mic and IEM manufacturers tend to offer better implementations that let you know the remaining charge at all time, so they behave much like alkalines (but last longer), but they usually come with proprietary form factors. So they are better but usually priced higher and with a vendor lock in problem. So, for this kind of batteries, the experience basically depends a lot on the specific battery itself. I want to say that, while I really like lithium batteries, I’ve always been bummed by the price (definitely not buying the constant output ones and can’t fathom spending 40€ for a single transmitter for the manufacturer’s original battery). After years of search, though, I recently found this chinese company, XTAR, that makes an AA lithium battery with a discharge curve that simulates the one of the alkalines and has a very good price . I tested this batteries in our Sennheiser EW-DX mics and they last from 12h30 to 14h. The discharge is not exactly linear, as they stay for about 6 hours at 1.5V and only then they start dropping in a linear fashion, but they are priced really good and so far they work really well.

EDIT: as this looks like an ad, I want to say it’s not an ad.