They're all grammatically correct.
1. You should listen to what people say.
This is a piece of general advice. It recommends that we should listen to others -- presumably to their wisdom, observations, reflections, advice, corrections, etc.
- It's "the wisdom that old folks share with us" and "the feedback from the community of our fellow man."
2. You should listen to what people have to say.
This one rather stresses the "have to say" element, so it hints at an urgency (or an umburdening) in what the people have to say -- such as the input they want to give at town hall meetings, or what the regular folks think about the antics of politicians, or the quality of television broadcasting, or the wholesomeness of movies aimed at teenagers.
- It's "what people have to say about a topic."
3. You should listen to what people are saying.
This one suggests that people are talking about you, but you didn't care enough to notice. It suggests a scandalized population, or at least a critical one.
- It's "what people are saying about you."
- This is what people (the folk) say:
"Life is short, so make the most of it."
- This is what people (the body politic) have to say:
"We're mad as hell and we're not going to take it any longer."
- This is what people (the community) are saying:
"Bonnie Parker is sure to come to a bad end if she takes up with that Clyde Barrow."
However, I think there's only a trend for the meanings to group up that way; those uses are far from fixed.
All we are saying Is "Give peace a chance."
YouTube - Give Peace A Chance - John Lennon
This famous lyric uses the expression "what we are saying" in the second sense, not the third.
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