There have been many hardware vendors who sold products that need constant remote support to be operational, and those vendors went out of business, or just decided to end support for the product. Sometimes this results in the product being bricked after a period of time. Sometimes the product can operate in some more limited way without hearing from HQ. I would hope that would be the case for a Tesla, though some things like traffic data, cloud-based navigation etc. would cease to work, just as they do when you go out of the cell network. If anybody has taken a Tesla for a long time out into the country away from all connectivity, they could find out if, after a long period of time, something fails. No updates of course.
Ideally, companies who sell products of this sort should put their source code and other things into escrow, to be released to a buyer or the open source community if the company dies. But few do.
As others note, some other company might take over the task and buy those assets. It would probably start charging a higher monthly fee for basic connectivity, probably more than the $10 you pay for premium connectivity. Now that other companies can charge at SC, somebody would keep them operating, possibly at higher prices.
Software updates might be much less likely.
