Assuming I follow the steps in this answer to deploy a new version of an application without downtimes. When I route the users to the new application server instance, the application will not remember their session so they will be logged out without warning.
Is there a way to keep the sessions alive over multiple servers or to make a "session snapshot" or similar?
In my case it's a Java application if that matters.
Store the session data on a separate server that both application servers have access to.
Store the session in Memcache, and allow both servers to have access to the memcache backend.
You could store them on a shared disk, but that's slow and inefficient. Memcache is pretty much the right way to do this.
Look into clustering support for your server. Properly configured clustering will replicate your session data between the two servers.
