Simple question, are there performance gains or any other benefits to make it worth the effort of updating the server?
Short yes! You can find a detailed Benchmark on http://www.lornajane.net/posts/2014/php-5-6-benchmarks. If possible i would try to update from PHP 5.4.40 direct to PHP 7. The performance from PHP 5.4 to PHP 7 is massiv. A change to PHP7 made our Craft installation twice as fast. You can find some further information here -> https://pixelandtonic.com/blog/php-7