The fan was making a loud screeching sound a few months ago, but that subsided. Today the Airport Utility App alerted me that the Airport Extreme thought it was overheating. I felt the outside of the case, and it was warm, but not hot. I figured it had to be the fan.
I snapped off the connector for the LED. The guide did say that they could break, and well, it did. I managed to solder on some leads to the board and connect up to the LED cable, but when the connector snapped, something must be broken in the PCB because the the orange LED will light up, but the green one will not.
I took the fan casing apart, lubricated the shaft, put everything back together and now the fan spins, the Airport Extreme doesn't think it's overheating, and aside from the lack of a green LED, things seem to be working correctly.
Be really careful with those connectors. They come off perpendicular to the wires, and it's easy to accidentally pry at the female connector that's attached to the PCB instead of the male connector that's seated in it. After breaking the LED connector I was a lot more careful and actually pried against the wires themselves instead of the plastic plug.