Minifying HTML

You have to be careful when removing stuff from HTML as it's a fragile language. Depending on how your pages are coded some of that whitespace might be more significant; also if you have CSS styles such as white-space: pre then you may need to keep the whitespace. Plus there are numerous browser bugs, etc, and basically every character in an HTML file might be there to satisfy some requirement or appease some browser.

In my opinion your best bet is to design the pages well with CSS techniques (I was recently able to take an important page on the site I work for and reduce it's size by 50% just by recoding it using CSS instead of tables and nested style="..." attributes). Then, use GZip to reduce the size of your pages for browsers that understand gzip. This will save bandwidth while preserving the structure of the html.

Sometimes, depending on the enclosing tags and/or on the CSS, whitespace may be significant.

Outside of HTML Tidy/removing white space as the other answers mentioned, there isn't much.

This is more of a manual task pulling out style attributes into CSS (hopefully you're not using FONT tags, etc.), using fewer tags and attributes where possible (like not embedding <strong> tags in an element but using CSS to make the whole element font-weight: bold, unless of course it makes semantic sense to use >strong<), etc.