May I suggest giving a little detail to the results of the the troubleshooting steps suggested in the Fix printing problems in Firefox.

In order to troubleshoot what might be causing the printing from stopping, we might be able to help replicate the issue you are seeing.

Another troubleshooting step that is done often is creating a new Firefox profile to see if any of the settings have become corrupt: Profile Manager - Create, remove or switch Firefox profiles

What happens when you start Firefox with a new profile?

After performing the troubleshooting steps the problem remains. I took the extra step of creating a new profile, but I still can't print. When I check the Ricoh error log, it tells me that an authentication error has occurred, which I don't get with any other application. I will contact Ricoh in case they've seen this before, but I can't help but think it's related to some change in a newer version of Firefox.

Is there a user permission to use Firefox on that computer that is different from other appplications?

The error you mentioned looks like its similar to the authentication of using the printer -http://support.ricoh.com/bb_v1oi/pub_.../0128.htm

However I am not sure how it would relate, if you right click on the Firefox icon and click on "Properties", are the user permissions the same as those of another application you print with (ie. a pdf reader)

I spoke with Ricoh, and it appears that Firefox is stripping a user code that we use to distinguish faculty print jobs from students'. I have attached a screenshot of the print screen to show what I mean. This user code is used for all print requests, in all apps, and works fine (and worked fine in Firefox until recently). When I send a print request from Firefox v.50.0, the printer is engaged (you can hear it click on), but then the job is cancelled because it does not receive the necessary code.