Let’s face it no one can really list every potential show stopping error where search engines are concerned.
The safest thing to do in that case is fix all your errors or as many as realistically possible.
The reality is as long as your site and pages display like they’re supposed to your visitors aren’t going to care whether or not the code behind your site is 100% valid.
Browsers won’t care much either as they’ll typically render much invalid code perfectly fine. Still there are good reasons to validate your code.
Validating your code is simply comparing it to W3C standards.
100% valid code means 0 errors, but it could include any number of warnings.
Some of those warnings aren’t going to be things you’ll want to fix.
Many tracking parameters you add to the end of a url will give a warning for example.
The home pages of Google (35), Yahoo (167), and Bing (14) all have validation errors.
However all things being equal I would think valid code better for search engines than invalid code.
One advantage of HTML Tidy is using an extension you can check your pages directly in the browser without having to visit one of the validators.