To paraphrase the American General, Douglas MacArthur, some older airports that are being replaced with new ones not only don't die; they don't fade away, either. Recent evidence of this was the announcement by ANA of Japan (a country General MacArthur had some experience with, by the way) that the airline was considering utilizing its new Boeing 787s on international routes from Tokyo's Haneda Airport, and not Narita, following initial domestic services with the aircraft type.
Prior to 1978, Haneda was Tokyo's sole commercial airport. When Narita opened in that year, all international flights (with the exception, for political reasons, of Taiwan's China Airlines) were moved to the new facility, making Haneda essentially a domestic-only facility. Due to the small number of slots resulting from the initial (and many years after, due to disputes with local land owners) use of a single runway at NRT, it was unusual in the sense that almost all of its flights were international; Japanese carriers ANA and JAL were unable to mount a fully-intregated hub from their home base as a result.
Another internationally-oriented example was Montreal's Mirabel, which also was designated as the facility for long-haul services (with a modest amount of domestic connecting flights, a la Narita), while Dorval, which previously hosted all of Montreal's airline service continued to serve domestic and U.S.-transborder routes. Air Canada still had a full-scale hub at Toronto, to handle most international connecting services, although the use of YMX undoubtedly diverted connecting traffic that had previously transited YUL (Dorval's code) away from the Montreal area.
In a 'back to the future' move in the late 1990s, international passenger flights were returned to Dorval, leaving cargo as the principal inhabitant of Mirabel. Thus, Air Canada regained a second significant location to flow transatlantic traffic to and from North American points, while travelers to and from Europe benefitted from Dorval's relative proximity to downtown Montreal compared to Mirabel.
Two relatively historic U.S. airports, Chicago's Midway, and Love Field in Dallas, have developed 'second careers', following the opening of new, larger facilities in their metropolitan areas. MDW was supplanted by O'Hare at the start of the jet age, since Midway could not accommodate the early jets. During the 1970s, the City of Chicago attempted to restore significant service on its South Side, by letting the U.S. Local Service carriers expand into the city via the older airport, as well as encouraging a modest amount of relatively short-haul service by trunk carriers such as American, Delta, TWA and home-town carrier United (whose headquarters had been located on Cicero Avenue, opposite Midway's Terminal).
Today, Midway, after being the home of one of the first post-deregulation 'new entrant' carriers ( the eponymous Midway Airlines, who also utilized the former UAL HQ building), is dominated by Southwest. The carrier also figures strongly in the continued use of Love Field for airline service. DAL was supposed to have become a general aviation-only field when the nearby giant DFW airport, serving both Dallas and Ft. Worth, opened in 1974, and all of its historic inhabitants duly made the trek to the new location. Southwest, however, had been established following the agreement to do this, and stayed at DAL. Legal battles ensued, with the result that, until 2014, services at Love Field continue to have some geographic and/or aircraft size restrictions.
Two prominent airports that did die were Denver's Stapleton, and Hong Kong's Kai Tak. Both are now totally bereft of aviation, and their land is being re-developed for other uses, including significant residential development at Stapleton. Their airport designator codes were transferred to their successors, Denver International and Chek Lap Kok, respectively. The signature curved approach to runway 13 at HKG is no more, probably to the relief of both many air crews and the residents under the approach.
A more exotic set of circumstances has occurred at the Cincinnati/Northern Kentucky International Airport, which once featured both passenger (Delta) and cargo (DHL) hubs. After DHL acquired Airborne Express, it chose to consolidate at the latter's hub at its private airport near Wilmington, Ohio, north of Cincinnati. Following the merger of Delta and Northwest, DL has chosen to downsize the CVG hub, including closing the former terminal used exclusively by its regional partners. DHL, meanwhile, has downsized its U.S. domestic operations, but has elected to move its remaining hub operation back to CVG.
And finally, the Japanese carriers once again are beginning to have significant international operations at HND, where they can be fed by their extensive domestic services. Interestingly, Haneda now has four full-length runways, compared with a total of two, one of which is only 2500 meters in length (and thus, not well-suited to takeoffs of long-haul flights), at its newer neighbor.