You should hand write your references on your resume.

This can be taken to mean a couple of things.
Overall - no - nothing on your resume should be written in by hand. If it’s going to be included on your resume it needs to be typed in and turned in at the same time as that resume. However, it’s not unusual for people to submit their references secondary to the initial submission of their resume. In that case what you can do is list a section on your resume that’s called References and underneath state “References available by request.” If they are then requested later, turn them in typed neatly on a document that matches your resume.