This was asked by a friend before investing in bitcoin. His fear was that a new technology would emerge to replace the internet, and that would make Bitcoin obsolete.
My answer: nothing would happen.
Bitcoin runs on the internet, but it does not depend on the internet. The fact that most of the nodes use the internet to communicate does not mean that the internet is necessary for Bitcoin to work.
As long as the nodes can forward transactions and blocks to each other, the Bitcoin protocol will continue to work. It does not matter which means of communication is used. The Internet is one of them, but there could be others.
The reality is that Bitcoin does not depend on or compete with the internet. Bitcoin is a value transmission protocol. The Internet is a data transmission protocol.
If there were a better method of data transmission than the Internet, Bitcoin clients (nodes) would gradually adopt it. And the technology that would become obsolete would be the internet, not Bitcoin.