VS Code Icon not updated in Big Sur I'm on macOS 11.2.3 (20D91). I updated VS Code to the latest version which introduces a new icon.
Unfortunately, icon remains the same of before. I have already run killall Dock and killall Finder but things did not change.
Fixed thanks to this suggestion
From the link above:

Restart your dock via killall Dock in the terminal

Run these scripts in your terminal (which updates the timestamp of the icons):
sudo touch /Applications/Visual\ Studio\ Code.app
sudo touch /Applications/Visual\ Studio\ Code.app/Contents/Info.plist
sudo touch /Applications/Visual\ Studio\ Code.app/Contents/Resources/Code.icns
killall Dock Finder

Note: For Insiders and Explorations, simply append \ -\ Insiders or \ -\ Exploration to each version name reference.

Force the IconService cache to be rebuilt using these steps:

Remove the main IconServices cache store:
sudo rm -rfv /Library/Caches/com.apple.iconservices.store


Remove the subsidiary store, which includes the Dock icon cache too:
sudo find /private/var/folders/ \( -name com.apple.dock.iconcache -or -name com.apple.iconservices \) -exec rm -rfv {} \;


Restart and give your Mac plenty of time to rebuild the caches.