Why is my airport extreme fan humming?

Noisy fans are quite common on AirPort Extremes that have been in service for 12-18 months or longer. The fan is usually inaudible, but if it becomes clogged with dust......it loses its cooling efficiency and must run at higher speeds to try to keep the electronics cool. As you might imagine as the fan speed increases, so does the noise and vibration. Soon, the fan will not be able to keep the electronics at a normal operating temperature and you will likely see messages to the effect that the AirPort might be overheating.

If the AirPort is less than a year old, ask Apple to replace the device for you. Another option if the AirPort is less than three years old.....AND....you have AppleCare in effect on a Mac, then Apple will normally replace the AirPort under the AppleCare umbrella. Get with Apple Support if you think the AirPort might quality.

Otherwise if you are a good do-it-yourself type person, you can open up the AirPort Extreme and replace the fan. Parts are available from a number of sources if you Google "AirPort Extreme fan".

The article linked just below might provide some tips on opening up the AirPort to pull the electronics.

AirPort Extreme A1521 Teardown - iFixithttps://www.ifixit.com/Teardown/AirPort+Extreme+A1521+Teardown/15044

A service shop could likely handle the repairs as well, but the labor cost might seem prohibitive.

Since you have already owned the Extreme for a few years.. several according to what you posted.. they were introduced mid-2013 so if you bought it early in the piece it is already 5 years old.. it is well and truly time to replace. Even attempting to fix it at this stage may not offer you a lot more life.. and it is as noted.. not a job I enjoyed at all.. particularly getting the connectors back into their sockets. Very fiddly and very un-Apple type engineering.. they built it into the wrong shaped box.

I would buy an alternative.. e.g. a really good one for Apple environment is Synology RT2600AC.. not cheap but a very accomplished unit. Higher end Asus have also been pretty good for me.. but they are not as robust.. although I prefer the layout of the setup.

Once you install and have the new router up and running.. power on the Extreme with the reset held in for 10sec until front LED flashes rapidly.. that will wipe all your values. Then sell it on eBay as working but needing fan replacement. Lots of people will want it for parts and you can get reasonable prices for them.

It won't hurt to try, but the fan is located up near the top of the AirPort, so getting air from the inlets at the bottom of the AirPort to the area where it is really needed is quite a challenge.

It never worked for me, but you might have better luck.

If you decide to try this, be sure to power off the AirPort and let it cool down for an hour or so before you attempt to blow cool air into the device.

Is the prognosis, then, I can expect this router to eventually start overheating and then expire?

Yes unfortunately, pretty much due to poor product design by Apple. Maybe you have a friend who is good with projects like the fan replacement.