What changes after peer review but prior to final submission are acceptable for a conference paper? Suppose I write a paper and submit it to a conference. The paper gets some weak rejects and weak accepts and lots of criticism, but the conference is not that prestigious, so they accept it anyway.

Obviously, if I agree with the reviews, I will want to change the paper (which is in part what the reviews are for). But how far can I go with these changes? If the paper is computer science, for instance, I may easily run additional experiments, or slightly change the setup and re-run the existing experiments. If I submit the paper with these results, I will basically be publishing experiments that have not, in their current form, been peer-reviewed.

At what point do changes to a submitted paper post peer review become unacceptable?
Following the acceptance of a paper, I would not make any changes that go beyond "editorial"—that is, improving the grammar, or adding a recently published citation. These do not change the "technical" content of the paper.

Anything where you make edits that change the actual research or results presented in the paper should be presented to the editors of the journal or conference in question, with a request for guidance. The individuals in charge can then make a determination whether or not additional peer review is required.