I had this issue too.

Clicked VS Code icon and it jumps but doesn't do anything.

Tried running $> code in terminal, nothing happens.

Tried updating VS Code by copying the latest VS Code.app and replacing it in Applications, still not working.

Tried VS Code icon again and this time it said it was already running but is not responding.

That last step led me to finally go into Activity Monitor to try and close VS Code, but couldn't find it. After some thinking I came up with the idea to search for Electron since I know VS Code uses it. Found it, double clicked and clicked Quit, and tried opening VS Code and it finally opened.

TL;DR - Open Activity Monitor and quit Electron, then try opening VS Code again