Based on the community response, the D5100 firmware update did break compatibility with some third-party batteries. So if using inexpensive non-Nikon batteries is a priority, updating may cause problems.

In general, firmware updates are usually a good idea because they can fix bugs or improve reliability, but in this specific case the tradeoff is battery compatibility. If your camera is working well and you specifically want broad third-party battery support, avoiding the update may be the safer choice.

The answer also notes that some users worked around the issue with an unofficial patch, but use that at your own risk.

For batteries, the safest option is always genuine Nikon batteries for full compatibility. If you choose third-party batteries, buy from a reputable brand and seller, and confirm compatibility with your exact camera and firmware version before purchasing.