I can't take my eyes off of you is correct and idiomatic. We would not side "I can't take my eyes from you." I don't agree with your book that "off of" should always be replaced with from. They can mean the same thing: "he jumped off of the roof" means the same as "he jumped from the roof." But one is not better than the other in that example. But sometimes, "off of" sounds casual or slangy. Example: "He bought a car off of me" is okay but slangy, and "he bought a car from me" sounds more proper. So, it's a case by case thing. Finally,, "He cheated off of me" usually has to do with cheating during a test. For example, if a student sitting next to you looks at the answers you are writing, you could say "He cheated off of me." "He cheated on me" has to do with infidelity in romantic/sexual relationships . For example, when a husband has an affair with (has sex with) a woman who is not his wife, the wife can say "He cheated on me."

Two different prepositions in play here. I can’t take my eyes off of you. <-- "Off of" is correct, but the colloquial here is plural. How do you make of the sentence: He cheated off of me. <-- For this it would be "He cheated on me."