What operating systems are you considering for this?
If using a Windows OS and using something later than Vista then you shouldn't have a problem with many thousands of connections on a single machine. I've run tests (here: http://www.lenholgate.com/archives/000568.html) with a low spec Windows Server 2003 machine and easily achieved more than 70,000 active TCP connections. Some of the resource limits that affect the number of connections possible have been lifted considerably on Vista (see here: http://www.lenholgate.com/archives/000837.html) and so you could probably achieve your goal with a small cluster of machines. I don't know what you'd need in front of those to route the connections.
Windows provides a facility called I/O Completion Ports (see: http://msdn.microsoft.com/en-us/magazine/cc302334.aspx) which allow you to service many thousands of concurrent connections with very few threads (I was running tests yesterday with 5000 connections saturating a link to a server with 2 threads to process the I/O...). Thus the basic architecture is very scalable.
If you want to run some tests then I have some freely available tools on my blog that allow you to thrash a simple echo server using many thousands of connections (1) and (2) and some free code which you could use to get you started (3)
The second part of your question, from your comments, is more tricky. If the client's IP address keeps changing and there's nothing between you and them that is providing NAT to give you a consistent IP address then their connections will, no doubt, be terminated and need to be re-established. If the clients detect this connection tear down when their IP address changes then they can reconnect to the server, if they can't then I would suggest that the clients need to poll the server every so often so that they can detect the connection loss and reconnect. There's nothing the server can do here as it can't predict the new IP address and it will discover that the old connection has failed when it tries to send data.
And remember, your problems are only just beginning once you get your system to scale to this level...