Well, seeing as a few companies tried to sue Sony for replacing several of its competitor's logos in time square in Spiderman (notably, Samsung Electronics, a Sony rival), it seems unlikely that someone could do the exact opposite (though obviously not impossible).
Article on the ruling from the Spiderman case.
As I understand it, there is some legal difference between having a product in a shot, and having an advertisment or logo appear in the background of a location. A product bearing a logo (such as the aforementioned bag of fertilizer) can easily be moved, covered up, or replaced with a 'fake' branded product.
However, companies make a conscious effort to have their brand logos prominently displayed in public areas (i.e. Times Square). They cannont in one hand justify spending a hefty sum to have their logo displayed in such a famous public space, and then on the other hand, sue a poor indie production because the companies logo appears in the background.
I'm sure you'd get in much more trouble trying to scale a billboard and cover up a 'Sony' logo with gaffing tape.
The only other option would be to censor your shots and make sure no logos creep in (as you said, an impossibility in Times Square), and THAT would be a first amendment issue.
The best option with products, even on extremely low budgets, is to just design something quickly yourself and print it off down at Kinkos. You can even glue it on to an existing box, of say cereal or something. Plus you can print it on a nice matte paper so theres no annoying shine from the lights.