Does your website use a print style sheet that specifies different fonts than your main style sheet?

If that isn't relevant, could you provide a URL for the page that has the problem (assuming it's public)?

Sorry, it is not public. But you can see the issue on any website using Font Awesome, even the official website: http://fontawesome.io/icons/

I've attached a screenshot from the print preview of that page.

As for your first question, the print section of my stylesheet on my site doesn't specify fonts. I tried adding the font specific styles to the print section without any change. I also tried removing the print section altogether.

Hmmm, strange. I tested in a new profile (no custom settings or extensions) and the official website previews just fine -- with respect to the custom font, at least. (Screenshot attached.)

Could you try Firefox's Safe Mode? That's a standard diagnostic tool to bypass interference by extensions (and some custom settings). More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using

Help > Restart with Add-ons Disabled

In the dialog, click "Start in Safe Mode" (not Reset)

Any difference?

I take it from your screenshot that you are using Firefox 25.0.1. The problem I am having is with Firefox ESR 17.0.4.

Safe mode changes nothing for me.

Oops. BTW, is there any reason you aren't using 17.09?

I can't test that immediately, but any volunteers wanting to try it can use the Portable Edition of Firefox 17 ESR from here:

PortableApps.com: Mozilla Firefox, Portable Ed. at SourceForge.net

Because the group policy at the place where I work is a nightmare.

The latest Firefox 17 ESR version is 17.0.11 and that also gives this issue for me on Linux, so I can confirm that a print preview with a downloadable font like Font Awesome doesn't work.
The Firefox 24.1.1 ESR version works and shows the Font Awesome in the print preview.