Does the FAA require the airlines to maintain continual communication with aircraft that can't be switched off?
No, there is no such requirement for radios, they can be switched off, they can fail, and propagation conditions may not allow to establish a radio link using terrestrial networks.
Every 10 seconds, the aircraft should report very basic parameters like position, altitude, speed, heading, engine power, and fuel. You can put all that into a single IP packet. Surely this must have occurred to someone? Are the airlines resisting
At the time of MH370 it was already possible using satcom and ACARS. ACARS can report a status using VHF or satcom. Satcom is used when overflying areas without VHF ground stations (desert, ocean).
MH370 had a satcom equipement to use ACARS, but didn't pay for the subscription, so they were not able to transmit ACARS when remote of VHF stations.
Why the subscription wasn't paid is likely because this would increase the ticket price, something passengers don't want, so airlines are indeed reluctant.
Nevertheless, considering the huge cost of Search and Rescue operations for MH370 and AF447, States are asking for the generalization of a space ATC. In space ATC communications with ATC can be done by multiple means, including satellites.
Space ATC is currently under preparation, but some elements are already deployed, e.g. the position can be reported from nearly everywhere using space-based ADS-B:
Aircraft equipped with ADS-B can stream their exact position and altitude twice per second, wherever they are in the world, to a network of 66 satellites.
Source: Space-based ADS-B helps NAV CANADA reach new safety milestone.
This question is recurrent under multiple forms, focusing on different aspects, so this overview is already refined in:
Why don't all commercial aircraft transmit GPS data in real time?
What are the recent developments in the satellite-based tracking of aircraft?
Post MH370, are there developments to improve the tracking of airliners? (closed as duplicate)
Live streaming cockpit to air traffic controllers (closed as duplicate)
Would better location tracking for airliners improve safety?