Get ID of featured image using “get_post_thumbnail_id(the_ID())” – without printing to screen?

You should not use the_ID() in this case, since it will echo the ID, use instead get_the_ID() to return it.

So please try this instead:

$thumb_id = get_post_thumbnail_id( get_the_ID() );

to get the ID of a post thumbnail image.

The general rule is that the_*() functions will echo the output, but get_*() functions will return it. But of course there are exceptions to every rule, so be careful 😉 If in doubt test it, consult the Codex or check out the source code.