tl;dr. Do not sign the document. Beyond this case, it could also affect your future employability. VCs fund good teams, not ideas, which are cheap. If your partner doesn't realize that, she also has a hard lesson in front of her. Another great HN post:
https://news.ycombinator.com/item?id=3844893+++++++
IANAL.* This is mostly based on my personal experience with NDAs, employment docs, and patent applications. NDA may also go by the name of a "confidentiality agreement" or a "proprietary information" document. You should also refuse to a non-compete agreement that says you can't pursue a related business or anything barring you from recruiting someone from this business. She may also ask you to sign a "work for hire" agreement that assigns your work product to her, which would also give away your ownership claim. Obviously, this work-for-hire contract is typically signed when you're getting paid as an employee or contractor.
More to beware: You may also be presented with documents assigning away all "inventions". Be especially wary of this, as you may be signing away your idea and your right to pursue something even vaguely related later. You may also be giving her what she needs to file a patent.
As others have suggested, company uses of NDAs are all over the place. Many companies do require them. Many don't. Strictness is all over the place, as is validity by state. For example, non-competes are almost unenforceable in California. Some companies who ask you to sign such an agreement will offer to compensate for the time you're under a non-compete -- for example, you can't work at a competitor for 2 years, so they pay your salary for 2 years after separation. If you had to sign a document like this to take a job, sometimes this stuff is negotiable, though not often at big companies or at the entry level. You may see "in perpetuity" given as the timeline for some of the things she's asking you to sign. Consider that as well.
The "unpaid intern" part may not even be legally valid. There are stringent employment rules governing interns, generally an internship is supposed to benefit the intern more than the mentor and it is generally illegal to just use interns as a cheap/free alternative to paid labor. That said, this often goes enforced.
Again, don't sign anything. It's better to be in an uncomfortable position where you haven't signed than in a position where you have signed and have to hire a lawyer to even get back to where you started. You need to think about future employers. They may ask you to sign general HR documents or NDAs, and one of the things they often ask you to legally acknowledge is that you aren't bound by other documents you've signed. This is why I really hate these things. Lawyers can't read code, and engineers can rarely understand the convoluted nonsense of a document than can come back to bite them later. You don't know who's going to be running the company 10 years later, when you're long gone but the documents you signed are still in force.
As sjg007 suggested, you may consider going to the professor.
Finally, the cliche is that ideas are cheap. (If she doesn't get that, send her this Onion video: http://www.youtube.com/watch?v=DkGMY63FF3Q) The reality your partner is going to learn is that even if you signed 100% of the code away, it's not much good to her. She can't maintain it and iterate it to a winning product. Anyone who can is either going to want money, or more likely equity. If the idea is truly a good one, as soon as another good dev sees it, or hears her talk about it (and isn't under NDA) could rewrite it. The source is irrelevant, there will always be clones for good ideas. VCs don't expect a perfect product, but they will fund a team that they think can get to one. Teams without technical founders don't get funded.
(*though I spent considerable personal money to hire a human resources lawyer specializing in NDAs when a former employer presented me with a very lopsided NDA. I considered that a small expense to protect my career.)