OP: Do we HAVE to respect others’ beliefs?
You don’t HAVE to do anything. You don’t even HAVE to obey the law. But you have to be prepared to accept the consequences of your actions.
As for respecting others’ religious beliefs:
I’m a hard-core atheist. I know that God and the afterlife don’t exist just as certainly as I know that Santa Claus and the Easter Bunny don’t exist. I don’t try to convert others to my ways, but I know what I know. Meantime, my live-in girlfriend is a devout Protestant. She is very set in her beliefs, and I doubt if I could argue her out of her religion even if I tried for years on end. We’ve basically agreed not to push our views on each other. We love each other very much and we expect to get married in the future and live the rest of our lives together. (Incidentally, my ex-wife was a Catholic–in other words, this isn’t a new situation for me.)
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Do I respect my girlfriend’s beliefs? No. Her religious beliefs are wrong. Plain and simple. Basically, this is just another way of saying that I disagree with her on this point.
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Do I respect my girlfriend? Of course. There is much more to her than her religious beliefs. She is worthy of respect in a thousand ways. The fact that I don’t respect her beliefs doesn’t affect my respect for her as a person in any material way.
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Do I show respect for my girlfriend’s beliefs? Yes. Call it paying lip service, but I respect my girlfriend, so I at least make a show of respect if she talks about some aspect of her religious beliefs. She does the same when I discuss some aspect of my own atheism. We know we disagree on these things. We don’t feel a need to be openly scornful of each other’s beliefs.
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Do I respect my girlfriend’s viewpoint on her religion? Yes. That is, I know about her past and her background, and I know that being religious makes sense for her. It doesn’t do any particular harm to her or me or anyone else, and I respect the fact that that religion fills a need for her. I don’t respect the beliefs themselves, but I respect how those beliefs complement her life and how she applies them in her life. I respect the interaction between her as a person and her beliefs.
The reason I’m spelling all this out is to demonstrate that a lot of different things are meant by the word “respect.” We all have family members and close friends who look at the world differently from us (who have different political opinions, who have different priorities in life, or who even just root for a different football team). And we can love the person, disagree completely with them on this or that point, and still respect (or at least show respect) for the difference–or, at a minimum, the other person’s right to hold that belief.
Co-workers are a little more remote from us than family and friends, so we may be a little less forgiving or respectful. But there are still conventions regarding how we deal with co-workers too.
People wearing tinfoil hats are the most remote, and we may feel the least restraint when it comes to the need to respect them or their views. But even then we get into political issues of “respect.”
“Political respect” (to coin an awkward phrase), means to me that I have to respect and even defend the rights of others to believe and even live differently from me, even in ways that directly contradict my own beliefs. I do it on the basis of a political principle embodied by the First Amendment and the general ideas of the Constitution. Basically, the health of the country is at issue at this particular level. Compromises have to be made, of course–members of the erty-erty religion can’t deflower 12-year-old girls whenever they want. But if the erty-erts don’t act in a way that infringes on the basic rights of others or harm the state, then I can’t attempt to restrict their ability to worship and live as they please. I can hate them, rail against them, and try to convert them, but under the spirit of the First Amendment I also should be fighting anyone who would attempt to restrict their right to practice the erty-erty religion. I have to respect their religious beliefs on at least THAT level.
Okay. End of my dissertation. Mainly I wanted to point out that there are lots of nuances to the word “respect.” The definitions I have offered and the differentiations I have pointed out aren’t authoritative or exhaustive, of course. Other posters can certainly refine and/or refute some of the nuances I’ve tried to explain here. But to answer the OP’s questions, I think each person need only look at their own lives: How does each of you react to those family members and close friends who have beliefs, priorities, or opinions that are different from yours?