Not automatically but when you open a dead link you can ease the pain by using an extension such as "page archive" [1] to quickly search an archive such as the wayback machine and perhaps get something useful out of it.

I've always wished a browser such Firefox would extend bookmarks as an offline archive of all links you add to it. Your own personal wayback machine perhaps.

[1] https://addons.mozilla.org/firefox/addon/view-page-archive/