In order to publish and remote debug your dotnet application you have to add appropriate entries in the vscode user/workspace settings.

For example:

{
    "dotnetDebug.remoteHost": "[email protected]",
    "dotnetDebug.remoteFolder": "/home/dotnet"
}

You can open the user settings.json file with the Preferences: Open User/Workspace Settings (JSON) command in the Command Palette (Ctrl+Shift+P).