Veeam Backup & Replication Console settings are stored per user in the file:

%localappdata%\Veeam_Software_Group_GmbH\veeam.backup.shell.exe_Url_hu1utqnj52thvmhrg5kie2bl15o22i22\12.0.0.0\user.config

The folder names within the "Veeam_Software_Group_GmbH" folder may change between versions.

 

At the bottom of the user.config file will be a section named  <Veeam.Backup.UI.Launcher.Properties.Settings>

To remove a server from the Veeam Backup & Replication Launcher drop-down box, remove its entry from the BackupServersList values in the user.config file.

Example (before):

        <Veeam.Backup.UI.Launcher.Properties.Settings>
            <setting name="BackupServersList" serializeAs="String">
                <value>VeeamBackupServer:localhost:9392:current;VeeamBackupServer:oldvbr:9392:current;VeeamBackupServer:tempvbr:9392:current;VeeamBackupServer:VBR:9392:current</value>
            </setting>

Example (after):

        <Veeam.Backup.UI.Launcher.Properties.Settings>
            <setting name="BackupServersList" serializeAs="String">
                <value>VeeamBackupServer:VBR:9392:current</value>
            </setting>