Add theme support custom header will not hide text?

Chances are, the theme you are using does not take into account the Custom header settings when it puts in the Site Title and Description.

I would suggest (if you are comfortable in PHP) creating a child theme (as described here) for your theme, then copy header.php into your child theme and remove/alter the area where it puts in your Site Header.

For more great tips on WordPress basics, the same guy covers a lot in his 24 Days of WordPress series.