I've heard that if I want to be certain that my website will continue to be found on the search engines, that I should regularly resubmit my site? Does this apply to all search engines, including Google and Yahoo?
Once you website has been crawled by any crawler based search engine such as Google, Yahoo, and MSN, there is nothing to be gained from resubmitting your site to them. They will continue to revisit your site periodically for as long as your site is accessible. You never have to resubmit your site to them again. There are no penalties for resubmitting your site to them, you can use there URL submission mechanisms as often as you like, but to do so more than once is actually pointless.
There are other search engines and directories that periodically drop stale content/listings if they are not resubmitted. The reason for this, more than anything, is to limit the amount of storage required to run the service, but also to ensure only active websites are listed.
Be aware, some search directories will remove your listing if you submit your URL more than once.