Will upgrading PHP version affect my current site that running Drupal 6?
As reported in System requirements, Drupal 6 requires PHP 5.x, since support for PHP 4.x has been dropped by PHP maintainers; it is recommended PHP 5.2.5 or higher, but PHP 5.3.x and higher may produce errors or unexpected behavior.

So, yes, PHP 5.6.14 is going to affect your Drupal site, especially if you installed third-party modules or themes.

Bear in mind that Drupal 6 core code will not be changed, after Drupal 8 will get its first official and stable release. If they don't fix Drupal 6 code before, it will not get fixed. Once Drupal 6 code is not fixed, it is also probable third-party modules stop to get updates too.