There was a problem connecting to server "SERVER NAME" error keeps appearing after install Sonoma

I can think of two possible ways to define a server automount.

You added a login item

1: Go to Apple menu > System Settings > General > Login Items. At the top, there is a section called "Open at Login". By any chance, is the server share in the list? If so, select it and press the – button to remove it. I suspect this is not the case as these usually prompt once per login session and stay quiet after dismissal. Also, if there is an item there but you can't remove it...

You MDM admin enforced a login item

2: Go to Apple menu > System Settings > Privacy & Security > Profiles (all the way at the bottom). Look through your profiles. If you have a corporate managed device, your MDM managers may have delivered a managed login item profile. This profile tells your machine to mount the server share. The behavior you describe is inline with this type of enforced behavior. Ah, but now that you are not on the same network, you are unable to access the server. But the profile keeps trying. In this case, you will need to reach out to your MDM admins and ask that the mount profile be excluded from your device.

There is a 3rd possible way to do this but I feel it is unlikely because it is too much effort. However, you can look in /Library/LaunchAgents, /Library/LauchDaemons, and ~/Library/LaunchAgents to see if there is a custom plist that is enforcing a server mount. A launchd item could also repeatedly force the attempt. Again, this would require creating the launch script and likely also an app to call. Decent amount of work but figured I would mention anyway.

Hope this helps resolve.

Reid