I recommend that first you should definitely check your site code is compatible with the PHP version you are thinking of upgrading to 7.0, 7.1 or 7.2 and the above WordPress plugin can do SOME (but not all) of the work. Unfortunately the only full way of knowing if your code is going to be compatible, is run it and see if you get any errors in the error_log under your theme or things break.
You could try switching PHP versions under staging first if you use staging.
The above article gives some general information about PHP versions (closer to the bottom of the article) that is helpful background to know what PHP 7.x offers feature wise.
Revert Back if Necessary
If necessary you can always revert back easily enough to PHP 5.6 if things break.
There were no performance gains that I could see on my site but already I'm using static page caching so what's served up is sent without backend PHP rendering of the page. If you are not doing that then in theory PHP 7 should speed up your site in general.
Good luck and backup before doing anything is an always wise approach.
Thanks to the Managed WordPress Hosting team for getting this done.