nostr:npub172v3nlrtzd8wssn8f3ckm47jtsxdhq0jakqua3crgrvn7mmv32vsju689n 
IDK about a burn button, but firefox has an option to clear site data when closed, you can add exceptions to that. I imagine you can do something similar with Vivaldi.

You can also containerize different sessions with the "Firefox Multi-Account Containers" extension to create a container, do the stuff you want to do, then delete the container.
https://addons.mozilla.org/en-US/firefox/addon/multi-account-containers/?utm_source=addons.mozilla.org&utm_medium=referral&utm_content=featured
 
I know this isn't exactly what you want, but hope it helps