Willravel

Skizziks has the right idea. These foods are incredibly cheap and redily available anywhere, but most importantly they're filling and get most of their weight from water (which I'm assuming isn't included in the $25). Lentils are extremely high in fiber and have plenty of protein. When combined with other cheap foods that can have other basic vitimans and minerals, you could survive without becoming sick.

If I were to do this, I'd also try and save seeds and try to grow food. Fruits like tomatoes are very easy to grow and grow relatively quickly. They're also very rich in nutrients. Rice, lentils, beans, and tomato along with maybe some pepper from a McDonalds could be dinner for a week.

The biggest problem would be meat. B12 is hard to find outside of meat products, even with the popularity of vegan diets. Milk may seem cheap, but on a budget of $25 it's actually quite expensive. The cheaper meats are probably going to be canned, but they're going to have more preservatives and less nutritional value. The best thing to do might be to get 1 turkey breast on sale and stretch it out over the month by only cooking a quarter of it a week and freezing the rest.

It would be very, very difficult. Even drinking a lot of water, I can imagine being quite hungry often over the month. What scares me, though, is that people forced economically to be on this diet are competing for food deals with other people on this diet. That means that all of the things we're thinking of in here are probably hard to find.