Does instant indexing remove pages that return a 404/403 response?
The instant indexing will remove 404/403 pages if the pages return an actual 404/403 response and not redirect to purpose built 404/403 webpage. Here are some further details in our how to set up instant indexing article
The normal crawl cycle (which is 3-7 days) should also remove the pages that return 404 response.
You also have the option to manually index the page using the diagnose tool which will remove the page instantly.
It is worth noting that an instant removal of a page from the index on a 404 response might actually not be preferable. This is because webmasters can often make mistake of using 404 where they should have used a 503 (a temporary service unavailability).