The main reason why errors are listed is that you can solve them (if necessary). If these are old pages that don't have existing links on your pages - you can just forget about these warnings. However, if these warnings appear because actual pages are linking to non-existing pages this will lead to a degraded user experience and user experience is a factor which counts for SEO.
If you look at the 403 errors - normally WMT lists how the bot got to these pages. If the pages that are linking to this 403 pages are still on your site, you have to remove these links.
If you have dropped in traffic, you could try to do a full crawl of your site using screaming frog of Xenu, to do a quick check-up of the technical health of your site.
If you still have an old sitemap, or the most popular pages in Google Analytics from the period before migration, you could also use these url's as input for Screamingfrog - and check if all pages were properly redirected. If errors pop-up, these would be the ones I would redirect. I understood from your initial question that the 403's where coming from very old pages which were never meant to be accessible.
rgds
Dirk