I take your question to mean: What problems might you have client-side when both products are installed?
Firefox doesn't rely on any features of Internet Explorer, and only on a very few features of Windows. So it's nicely contained in its own install area -- there is no Microsoft "DLL Hell" to consider. In the case of IE, you can damage IE if you install or uninstall dependant products like InfoPath. That generally doesn't happen with Firefox unless you are deliberately trying to be destructive. Firefox and Thunderbird contain a separate, complete install each. There are no shared files.
On the admin side, Firefox supports NTLM authentication out-of-the-box, so you get single sign on automatically. Windows user information has no particular obstacles.
If you really want to create a problem then have two users run separate versions of Firefox at the same time with the same Firefox profile on the same box. That doesn't work yet, but it's also quite hard to do.
This was first published in February 2005