Anvilicious: In case you didn't get it, Robert Heinlein dislikes mainstream religion (particularly televangelists), likes polyamory and also likes "pleasantly plump" women. He also advocates personal development (in a semi-spiritual sense). In this book, if your mind and spirit have been trained, you get to have superpowers.
Jubal Harshaw is a "LL.B., M.D., Sc.D., bon vivant, gourmet, sybarite, popular author extraordinary, neo-pessimist philosopher, devout agnostic, professional clown, amateur subversive, and parasite by choice." Yes, that's three doctorates. He is also incredibly wealthy, does little real work and keeps a harem of sexy secretaries. Oh, did we mention he just happens to be the Author Avatar?
Valentine Michael Smith has vast psychic abilities owing to his Martian upbringing. Among these is the ability to make any object, regardless of size or make-up, just "go away." This includes humans, and Smith spends most of the book with an odd mix of Blue and Orange Morality and Black and White Morality, meaning if he perceives a "wrongness" in you, you're just gone. Mike's bodycount, not that there are any bodies to count, gets up into the high hundreds.
As Jubal points out in the book, the moral code that religions always ascribe to their gods is not the same as that they insist on for mortals, and this combines in disquieting ways with Mike's philosophy that all people are God.
There is a second and more deliberate Nightmare Fuel in the notion that the "Old Ones" of Mars are capable of dismantling planets, have done it in the past, and simply haven't made up their minds yet on whether or not to do it to Earth.
A subtle, psychological example: You can learn telekinesis and other amazing psychic powers but, to do so, you must practice a discipline that will drastically shift your values and way of thinking. If you choose not to learn it, other people will. In the end, people who are hesitant will have to choose whether to give in or to live as a powerless throwback in a world full of superhumans.
Seinfeld Is Unfunny: This book wasn't written to exploit the counterculture movement of the 1960's, but predicted it. Since then, free love has become significantly less shocking and controversial (although something of a Dead Horse Trope). However, Heinlein's publisher pulled it off his "too controversial to print" list and released it to exploit the counter culture movement of the 1960s.
Jill's ideas on women, homosexuality, and sexism are frighteningly reactionary for such an otherwise progressive book. Her Character Development consists largely of the more liberal Jubal (and especially Mike) getting her to recognize this and become less prudish (she becomes a model in a carnival "posing show" to abandon her inhibitions). Duke undergoes a similar, if abbreviated arc, and the main conflict of the story is Mike's "church" running headlong into the reactionary values of society.
All of the male characters, even Jubal, talk patronizingly to Jill and the other female characters, and love to lecture them about why they're wrong. Note that Miriam is a scientist and Anne is a Fair Witness (a completely unbiased legal expert).
Spanking is mentioned or brought up with what Adam Cadre called a "what the fuck is your problem" frequency, which doesn't jibe well with the misogynist tendencies described above.
One of the book's main points is that women can and should enjoy sex (controversial back then, due to held-over Victorian beliefs). That's fine, but what is not fine is the language used by men to describe these pleasure-loving women. Michael sings a not-very-funny ballad about Jill as a "willing tart" and her friend Dawn as someone who "never shops for pants"; a Jewish man rather improbably greets his wife with the words "you limber Levantine whore".note This doesn't even make sense. They are in a group marriage, so she's not a whore.