I installed Live Share in VS Code but it is not showing me the login window, instead a tab for it appeared which has 2 buttons but they are not responding.
I ran into the same problem today. I'm not sure what causes this error but as suggested in a related VSCode issue I used the repair command that comes with the Live-Share extension:
Open VSCode's command prompt:
ctrl
+ shift
+ p
Enter the following command:
>Live Share: Repair Installation
Follow further instructions
This fixed the problem for me.