I'm downgrading node.js from v7.7.4 to v0.10.48 because I'm on a CentOS 6.5 server and apparently v0.10.48 is the latest it can handle. Are there issues with going backwards like this?
There are no issues in way that it will work just as well as it did when it was the current version.
There will be issues with the compatibility of modules on npm. Many of the modules may not work. That is partially because Node v0.10 uses V8 v3.14 (for comparison the current Node versions use V8 v5.x) so a lot of modern JavaScript syntax doesn't work there.
For details of what syntax works and what doesn't see:
For example you will not be able to use a simple arrow functions:
const add1 = x => x + 1;
or even the const keyword for that matter. You will miss a lot of syntax that we now take for granted even on the frontend. Remember that io.js was forked from Node v0.12 mainly because it used such an outdated version of V8 - and it was literally years ago and was about a newer and much modern version than the one you want to use.
Also the core modules changed during that time changed a lot - compare:
Because current versions of Node support newer language syntax and have improved core API, a lot of the modules on npm will not work correctly or at all. Make sure that whatever module you use still supports Node v0.10 or use an old version that did support it. You may need to use shrinkwrap to downgrade a lot of sub-dependencies because even if you use an old version of a module that used to work on Node v0.10, it may still load the latest versions of its dependencies that don't work any more.
Also there will be potential issues with bugs and vulnerabilities that will not get fixed. The v0.10 version manitenance ended on 2016-10-31 and the status of this version is currently: End-of-Life. The oldest version that is still maintained today (as of May, 2017) is 4.x.
See: