So once an Immutable Server is in place, nothing should change at all. Or is it only configuration.
Are source code updates acceptable on Immutable Servers?
Immutable is exactly what it means, immutable, no change on configuration or code running or system library or whatever, if a change has to be made, create a new image and deploy it, never change it while running.
Source code updates are the least thing to change on a running server, this should not happen on a running server, immutable or not.
You should create a versioned release of your code to deploy on servers, once you have this level of fixed versioning you can step ahead to versioning your server image which include this code release.
Any change, being a code patch or a system patch or a configuration change should trigger a new release to be deployed.