Reproducing the precise wording of a saying or the exact words someone has said or someone might say requires adherence to a simple set of rules of punctuation and capitalization, as described and demonstrated in the discussions following each of the examples provided below.
1. The old saying, “What gets rewarded gets done,” is applicable to any business process.
Setting off a saying, or a question or any other type of quotation, with commas marks the quoted material as the only specimen of that type of thing. Because this is not an attributed direct quotation, it should be presented with nonrestrictive construction, indicating that it is merely one of multiple possible sayings: “The old saying ‘What gets rewarded gets done’ is applicable to any business process.”
2. Demonstrators chanted “release the tape” and “we want the tape” as they marched down the street.
An attribution (an identification of one or more speakers) must be followed by—or preceded by—a comma (in the former case, a colon is sometimes used instead), and the first word of a full quotation should be capitalized: “Demonstrators chanted, ‘Release the tape!’ and ‘We want the tape!’ as they marched down the street.” (Notice, too, that exclamation points have been inserted at the end of each quotation to indicate that the speakers raised the volume of their voices above the normal range.)
3. When you tell young people to turn off the phone, they hear please cut off your left arm above the elbow.
When describing at a distance of time and/or space what a person or people say or would conceivably say, treat the statement as an actual quotation: “When you tell young people to turn off the phone, they hear, ‘Please cut off your left arm above the elbow.’”
@Mark: I don’t understand if you’re saying that you need “half-quotes” (what I think you’re calling nonrestrictive) within full quotes, which I think I was taught was the rulel, or to always use half-quotes for something like this, although I don’t see a real rule here. In your examples, the actual sentences (in bold) are not in full quotes, but when you offer the corrections, you have full quotes around the entire sentence with half-quotes around the supposedly quoted verbiage. For this reason, the first example didn’t appear incorrect to me when using full quotes around the motto, since the rest of the sentence was not in quotation marks, although the other two certainly needed your revisions. Can you expand on this a bit?
I agree, the three examples given for discussion did not have full quotation marks. Since the entire sentences are not directly attributed quotes, the corrected sentences appear confusing.