On the homepage and on section pages, articles with no images show up with "no image" icons that are different in every browser -- in IE, they look like little sheets of paper with rips in them.
How can I edit the HTML so that nothing shows up if there's no picture?
Can you specify which template set/theme you are using, default one? Usually, such problems occur when template tries to show image with specified number.
In your templates which are used for front and section pages, you can try searching for <img alt="..." src="{{ uri options="image 1" }}" /> (or "image 2", or whatever the number) and wrap it in
{{ if $gimme->article->has_image(1) }}<img alt="..." src="{{ uri options="image 1" }}" />{{ /if }}
Even better, you can show image even if you don't know it's number -