Why does the Minify Html option in Magento 2.1.2 break my (or other's) site?

I have had exactly the same problem as you was having.

When i made a production site i also tried enabling everything that had to do with caches as well as the html minification.

What you could do is checking the php error log on your webserver. There you can see exactly which file is causing you trouble. It probably has something to do with php tags like this:

<?php //samplecomment?> <--- this breaks it!
<?php //samplecomment ?> <--- this doesn't break it

You can read more about this here: https://github.com/magento/magento2/issues/5316