Removing headers from blog post "excerpts" (using Hacksaw)
You could use the Twig replace filter with a regular expression, this'll remove the tag and it's content.

{{ entry.body | replace('/&lt;h([1-6])&gt;.*?&lt;\\/h([1-6])&gt;/', '') | raw | hacksaw(chars='300',allow='&lt;p&gt;', append='...') }} 


The only limitation is regular expressions aren't fantastic with nested html tags, so your mileage my vary!

You might need to play around with the Regex a bit if you want to just limit it to a few headings, this will strip all h1, h2, h3, h4, h5, h6. You also might not need the | raw filter, it seemed to escape everything after the replace when I quickly tested it.