Is there any danger to Craft from upgrading from PHP 5.5 to PHP 5.7?
Don't worry, you should be fine.

Craft currently supports a minimum of PHP 5.3. Any higher version of PHP is also compatible.

When Craft 3 ships (late 2016 / early 2017), it will require a minimum of PHP 5.6.

Side note... If you're about to upgrade PHP, I highly recommend switching to PHP 7. The performance boost is notable, and should speed up your site by about 50-80%!