Typically, the MAC address is not public beyond the next router. Software could intentionally leak it, but I don't think that's likely on a system built for anonymity. However, it is possible that, should your traffic get traced to your true IP address, interested parties would attempt to then trace it to an individual. Any router you connect to could be storing access logs (or even passing them on to the next connection point) for a long enough time that they could narrow down which MAC the traffic came from. If they have you on camera, and the recorded MAC matches your PC, that's a bit more evidence. Sure, you could potentially fight it in court (MACs can be trivially spoofed, after all!), but why bother taking the risk?

It's not part of the IP packet, but in some previous cases exploits on Tor (such as the one the FBI used in the Freedom Hosting takedown) have explicitly queried the MAC address and then exfiltrated that information. I assume the intent was that they could then arrest the suspect and compare the captured MAC address to the physical machine to prove it was the same person.

In addition to what the others said, it could be used to correlate you across multiple e.g. public wifi hotspots. Imagine some dissident in a repressive regime leaking information and law enforcement checking the logs of the routers for various public places. If they find a MAC at all locations, they might be able to check security cameras and see what individuals were present at that time and correlate that MAC with an individual, and then further use that information to track their movements.

Yes it can be spoofed and someone could potentially be framed, but it's just more information that can be used in conjunction with other data to help deanonymize a person.

Have you noticed that Starbucks has wifi sponsored from Google? Considering that Google tracks everything else, it is reasonable to assume they track MAC addresses at nearly every Starbucks, too. It has been reported that shopping centers do this as well. You do not have to actually be connected to their SSID either because your MAC address will be broadcast with any frames transmitted.