Generally speaking, no.
As for the specific situation, New York Oceanic has been closed since midnight UTC:
DUE TO AN OPERATIONAL EMERGENCY NEW YORK AIR ROUTE TRAFFIC CONTROL CENTER IS DISCONTINUING AIR TRAFFIC SERVICES IN THE NEW YORK OCEANIC FIR UNTIL FURTHER NOTICE.FLIGHTS CHOOSING TO ENTER NEW YORK CENTER AIRSPACE SHOULD EXPECT ONLY ADVISORY SERVICE.
NOTAM A0249/20 refers.
Most ATS units have contingency procedures in place to dictate what happens if the unit needs to close on short notice, for example during an evacuation. It is quite common to have agreements with adjacent units to help clear the airspace ("clear-the-sky" systems). This is possible if adjacent units have RADAR and radio coverage that overlaps with that of the unit being evacuated. While not able to provide regular service, adjacent units can monitor traffic and make sure everyone gets out of the evacuated airspace safely, after which that piece of airspace is closed.
In order for an adjacent ATS unit to actually take over and provide normal service, several things would be required:
- Additional staffing. It might not be possible to just call more people to work, as that would mess up the entire roster in days and weeks to come, seriously affecting the non-evacuated unit.
- Additional working positions for the extra staff to actually sit and work, while still maintaining a number of backup positions in case of equipment failure.
- Staff must have knowledge of local rules and procedures of the airspace they are taking over, and have valid and current ratings to be allowed to control it. You can't keep your rating current if you don't use it regularily. Just like you can't just take an Airbus-pilot and ask them to fly a Boeing-aircraft because the Boeing crew called in sick.
- RADAR coverage must be sufficient. This could be achieved by sharing RADAR-data between facilities, but if one ATS unit needs to evacute, their technicians have probably evacuated too - which means the data cannot be used since there is no one to monitor the status of technical equipment.
- Radio coverage must be sufficient (see above)
- All telephone lines must be re-routed (which is doable, if the system is set up for it)
What is more likely to happen is that the unit evacuating will move into a contingency facility from where they are able to provide anything from a very limited to full service. Some airports have old control towers that are still being maintained as contingency positions, and area centres might have dedicated backup facilities, or possibly training facilities (simulator positions) that can be switched over to a live environment. Of course, if all the staff that happened to be on shift at the time of evacuation needs to be quarantined due to a contagious disease, you will have a pretty big staffing issue.