It's difficult to answer the "will it break" question because we don't know what extra modules, plugins, or themes you've installed on your Drupal or Moodle sites, and when you last updated them.
Core Moodle and Core Drupal are probably fine on PHP 5.4, 5.5 and 5.6. For some of your modules PHP 5.6 things might get a little breaky. PHP 7.0 will definitely break Moodle older than 3.0
Always always find a way to test as close to the real environment as you can. I use virtualbox on my PC to replicate the live environments. It gives me confidence that it will work out all right when I do do upgrades, and it also gives me the opportunity to fix things if there's something wrong. It really is worth investing the time and effort to test; you don't want to spend the weekend fixing things on the live environment.
Some hosting companies allow you to enable PHP modules through the .htaccess file. Check with your host.