The short answer is no, you can't get the IP address of a Facebook user via their profile or other means through Facebook. You'd need access to the Facebook server logs and without legal interjection, Facebook will not hand over those server logs to anyone. Even if you ask nicely...sorry.
The long answer is yes, but you would need to establish a direct connection to the user outside of Facebook. If they emailed you directly, you could try to trace the email.
Another option would be to connect directly with them through certain chat programs like Skype and use a Skype Resolver. I've run some tests with friends to try and find their IP through Facebook chat, and although there are folks that claim this can be done, I was not successful. As expected, I only saw the IP addresses that belonged to Facebook.
In summary, most people want the Internet Protocol address of a Facebook user because the user is harassing them. Don't reply to these messages. Simply block this person (usually a fake profile) and report them to Facebook. Case closed. If they're threatening you, report them to Facebook and strongly consider getting local law enforcement involved.