Airport elevations are given in MSL (Mean Sea Level). With the sea levels rising due to a warmer planet, I assume that elevations would need to be changed at some point. What is that point? A quick search says that sea levels have risen 6.5 inches since 1950 (3 inches in the last 20 years). It doesn’t appear that changing airport elevations is something that will happen anytime soon. But when? When sea level rises one foot since the elevation was established? Five feet? Somewhere in between?
Just guessing here, but since light aircraft reportedly have altimeters calibrated to 30,000 feet with a precision of ± 60 feet while commercial planes have altimeters calibrated to 50,000 feet, with a precision of ± 80 feet I’d guess they won’t be bothered by sea level variations until they’re greater than an altimeter’s margin of error.
The US Geodetic Survey has a vertical datum that was set in 1988, replacing one set in 1929:
https://www.ngs.noaa.gov/datums/vertical/north-american-vertical-datum-1988.shtml
I don’t know for sure but I expect that airport elevations are set based on this datum. So it’s up to the Geodetic Survey to update to a new datum.
I don’t think the altimeter error matters. The altimeter setting given to aircraft is calibrated so when the aircraft is on the ground at the airport reference point its altimeter reads the published altitude for that airport. The altimeter error then presents as a difference between the published altitude and the indicated altitude. You could decide today that the MSL reference is 100 feet lower, add 100 feet to the published reference altitude for an airport, give an altimeter setting to an aircraft calibrated for the new airport elevation, and the aircraft altimeter will happily display the new altitude.
And the altimeter is set according to air pressure which is constantly changing, so precision of a foot is basically irrelevant. It’s just important to know whether you are clearing the houses and powerlines (and that pesky hill) as you approach. If 6 inches - or even 6 feet - matters, then your problem is not the airport’s published altitude.
Some runways will have their elevations raised to stay above sea level.
I pulled a few of my old aeronautical charts from 2003 and compared them to the latest online charts. Here are the elevations of some airports listed then and now.
New York LaGuardia: 22’ / 21’
New York JFK: 13’ / 13’
Newark: 18’ / 17’
Dallas-Ft. Worth: 603’ / 606’
Wichita Falls - Sheppard AFB: 1019’ / 1019’
St. Louis Lambert: 604’ / 617’ 
St. Louis Spirit: 463’ / 463’
Belleville - Scott AFB: 459’ / 459’
If pilots can deal with a 13’ change in altitude over the last 20 years in St. Louis (did they change the measurement spot from one end of the airport to the other?) they can probably deal with a rise in MSL.
Well here’s an answer I can actually provide.
The airport elevation is the center of the airport which is calculated off of the highest elevation on the airport. That can change if a runway length changes, or one gets removed or added. In looking at the St Louis Lambert field they seem to have added the 11/29 runway and the elevation is at the end of 11 which explains why it went from 604 to 617.
I’m trying to find out if the elevation has to be on a runway or can be anywhere on the airport.