@dleroari Thanks for the response! I have thought for a few months now on whether it is worth the upgrade. In the end I have come to the fantastic decision of Maybe.
The biggest problem would be if I leave that codebase as it is, it will become more and more problematic to place newer apps on the same server (without moving to a fancier solution like multiple php versions). I am not a server administrator, so I try to keep server dealings at a minimum.
Our current setup is we have a 3 tier server architecture internally for apps we want access to on our network (the Intranet, employee directory, various forms, front-ends to microsoft databases, etc). We also have a database server (for hosting all of these databases) and a file server (for storing all of these files). The current desire (maybe I need to be convinced otherwise) is to have a single server (think: we don't have unlimited IP addresses, even internally. My networking team might dislike me if I add too many) where I can run all of the various apps for internal use. This makes server management a little more simple, and is less complicated for our network team.
So, this is the main reason for thinking it might actually be necessary. It hasn't been deemed necessary for the 3 years or so since it was built, but now presents problems. I wasn't thinking and built a Laravel 5.6 app on the same server and Laravel 5.2 by default didn't like PHP7.1 with how the app was configured. So I can foresee many problems in the future if I want to continue having a single server without multiple versions of php and such.
P.S. I have a cloned development server (literally a cloned vm) so I can test with the exact configuration of the server. And backups are running. Definitely important! I'll also likely do a git branch specifically for the framework upgrade, just to track changes separately and be able to support the live/dev versions while I do the upgrades.