In Settings/Reading I set Testimonials as post page.
How can I change the header of that page?
You could use a template. You just have to copy the file
header.php and call the new file with the slug of the page
testimonials. It would require 2 new files :
then you can call your custom header with
See the template hierarchy. Just be sure Testimonials is your front page.
EDIT: Regarding front page (1rst page of the site) WP looks into your theme and the first file to be called is
front-page.php. Most of the time WP blogs set front page as posts page (reading settings).
Here your can set another file as custom header and call it in
front-page.php it will be called on front page automatically.
You could also insert some conditional tags in the main
header.php but it's better to use
front-page.php when cutomizing front page.