Is there any danger to Craft from upgrading from PHP 5.5 to PHP 5.7? As of July 2016, PHP branch 5.5 is unsupported. The PHP version my server currently is running is 5.5.9 which implies that the server will not receive any security updates going forwards. 

I'm told that Craft recommends using 5.7. Is there any danger that parts of my website may stop working if the PHP is updated?
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%!