This question pertains to large towers at large airports. I'm assuming there is backup power in case of a power cut / power failure. But how long does a typical reserve generator / unit supply power for and what happens if it runs out without a main supply back online?
A nearby ATC facility I am familiar with has a backup generator which will run their ATC functions (TRACON and Tower), and is fueled to run for 4 to 5 days. After that time, it needs more fuel. After two weeks, it needs periodic maintenance (oil change, etc.).
The on airport approach radar has a separate generator at the radar site, which can run for about three days before needing refueling.
The airport has several backup generators. The ones which run the airport lighting are not capable of running all the airport lighting, and do not power the beacon which is at a perimeter of the airport and without backup power. I do not know how long the airport lighting can be operated.
Some of the NAVAIDs have migrated off backup power.
The terminal building has several generators, and I do not know what their duration or backup coverage is, but I would guess they can partially run the passenger terminal for a day or so.
Various FBOs, corporate hangers and freight operations have their own backup power. My employer has enough fuel and generator capacity to run for one week, prior to needing refueling. The A&Ps handle generator maintenance, but there is only one powerplant, so servicing, such as oil changes would happen most likely during daylight hours, by the A&Ps. I cannot address the other operations.
The FAA FSDO office is on backup power for the office.
One problem with backup power is that it does not assure backup communications. So in the case of businesses, they may loose connectivity. Some of the ATC communications have service levels to include communications even with widespread power outages.
In 1992, power was lost at much of the airport for 3 days or more, and some of the gensets were updated after that outage. My understanding is that even ATC backup power is not standardized for all facilities.
When the generators run out of fuel, they typically shutdown. Often longer outages are due to weather or natural events. In those cases, there may not be the ability to get a fuel truck in to deliver generator fuel.
Addendum: Talked with one of the tech who works on a company genset. He says that they do not use natural gas because the supply might be disturbed with earthquake action. They all use diesel. The company genset was trial run on Jet-A but diesel is cheaper, and they have contracts with two suppliers. If the tankers can get to the airport, the loaders can cut a path to the gensets (in snow). Gensets vary but generally expect to shutdown every 100 to 500 hours for fluid and filter change. For larger industrial backup gensets, filing the tanks can be done while running. The genset he oversees has a 10,000 USgal tank plus 500 gal in the genset housing. Systems for places like Alaska have more complex fuel temperature management to avoid fuel wax precipitation. Larger enterprises, like the regional hospital down the street, use multiple gensets to permit servicing, and to provide higher availability statistics.
Nearly all generators designed to serve as a backup power source in a safety critical environment (Hospitals, Air Traffic etc) will be designed to run 24/7 "indefinitely" with hot refuelling.
Obviously there will be a real world practical limit to that, but it will be many days if not even weeks potentially depending on the generator.
I suspect there will be huge variation in the type of standby equipment available, pre-loaded fuel and so on depending on the mission. For example, it would be entirely possible to have two generators which can switch over - allowing the other to be serviced and maintained.*
*NB: I don't know if this is an actual "thing", but I'd expect it in certain circumstances, for example, military.