When changing the IP-address of the ConfigMgr server it's needed to register certificates for the site server and boot media again. This is done by default during ConfigMgr installation and configuration, but not after changing the IP-address on the server. Just have a look in the ConfigMgr console, Administration, Security, Certificates.
Here you will see multiple certificates which are partly blocked and unblocked. Just open the unblocked ones, and install certificates when needed. In my case this was needed on all unblocked ones! Just install them for "Local Machine" and "Trusted Root Certification Authorities" in the wizard, and you will be fine after that.
After that communication on site systems and deployment is running fine again. No further issues after the IP-address change at all.