Most likely the MySQL was recently upgraded on your server. However, the server PHP is still utilizing the previous MySQL libraries, the MySQL version that was installed before the MySQL upgrade. Thus, there is a library mismatch.
Most likely the server PHP was compiled with --with-mysql=/path/to/mysql_config, an explicit instruction to tell the server PHP to utilize a MySQL config, an outdated compile method. You will want to upgrade the server PHP to the latest version using just the --with-mysql WITHOUT the explicit path (more information can be found here) or recompile the current PHP version again with the --with-mysql= pointing to the upgrade MySQL config.
PHP 7 Notes: (noted on Jan 10, 2016)
For those trying to upgrade the server PHP to PHP 7, there may be additional module that needs to be installed: "php7.0-mysql. You may also need to enable/re-enable the MySQLi extension. There has been few reports of PHP 7 and MYSQL connectivity. This is a server related issue. As long as PHP and MySQL are fully functional, PHP Live! will load correctly.
NOTE: After upgrading/recompiling the server PHP, if your PHP Live! system is indicating a "mysql extension is deprecated" error, please reference the "mysql extension documentation".