Does my fiancé have to leave the country to get a green card?

Yes, she will have to move out of the country.

She will not have to go back to her country if someone petitioned for her or her parents (Form I-130 or I-140 submitted to USCIS for her or her parents) on or before April 30, 2001. If someone (or a company) did that for her or her parents, then she does not have to return to her home home country to get her green card. If there was no prior petition filed before April 30, 2001, then she must be processed back in her home country for the green card and she cannot apply for it in the United States. I recommend not going through the process and have her wait for the Dream Act, which is much more of a sure thing than going through the US consulate or Embassy abroad for processing because she is likely to be refused. If she has US citizen spouse or children or parents who are really sick in the US then it may be worthwhile to process her case abroad with her husband's petition because the US consulate is more likely to approve her visa if she has to come back to the US to take care of sick family members who need her here.