You'd need to customize the F2F form for this. I'm going to blog about how this is done (it is indeed possible) at some point... if I ever get through the rest of my blogging backlog!
But do keep in mind that the more information you ask for from the referrer (the original recipient) the less likely they are to fill out that form. After all, filling out the form doesn't really reward them -- it's a shortcut (sort of) to just clicking "Forward" in their mail client. Also, the F2F model expects that the new lead (the referral) converts as a result of receiving the forwarded email. So you may want to personalize the email when it is sent to a friend (as opposed its normal appearance) to emphasize a CTA, etc. An easy way to do this is with a Velocity (Email Script) token:
#if( $lead.LeadSource == "Forward to Friend" ) Your friend likes you. #end