Should I remind potential mentors to look at my graduate application? An instruction email from one of the graduate schools I applied states the following:

"for the potential mentors you list on the application, after the January deadline passes, it’s a good idea to email them directly with a nudge to look at your application"

Is this necessary and is this a common practice?
It is not "necessary", but, if you are polite and considerate in your emails, at worst it won't hurt anything. Given the ease of application, most departments are deluged with applications, and it is easy to overlook (or forget to return to) any particular application.