Yes on a pragmatic basis, if the coworker is male and gets upset at being referred to by "he", but it goes against your own personal beliefs to refer to him as "she", it's best just to refer to him by name and practise wording your sentences to be pronounless. And, where possible and not disadvantageous, to avoid situations where he's involved in your own work.
With this approach, he's less likely to make a complaint to HR about you (though he might notice the careful lack of "she", but that's much more difficult to make a substantiated complaint about), and you still get to stick to your own beliefs.
It's still somewhat vexing to have to do this, but at least it prevents you from getting in the crosshairs of HR.
If it goes against your personal beliefs to call someone what they prefer to be called at no expense to yourself, then you need some new personal beliefs.
What about neopronouns? Can someone just make up a new set of words like zi / zim / zis and expect you to remember them whenever talking to or about that person?
You already remember what standard pronouns to use to refer to each person in your life. They're words you learned right from the start of learning the language. The same isn't true of "neopronouns".
Personally, agree with calling people what they want to be called. That said, here's a thought experiment: What if someone is inconvenienced? What if someone feels uncomfortable using pronouns that don't match the sex of the person? What about uncommon "neopronouns" like "zhe", "xe", or "fae"?
We expect people to say things that make them uncomfortable all the time. I don't feel comfortable telling my boss that I'm the one who wrote the buggy code that caused the incident, but I have a responsibility to do it regardless. I might be expected to thank everyone involved in a project, even if I don't feel personally grateful to them. And so on.
Obviously there's no easy way to reason these cases from first principles. As it is, I'm aware that being affirmed in their gender identity is recognized as therapeutically important for trans people. On the flip side, I'm not aware of any condition that causes people to suffer significant distress due to using a particular pronoun. So in this case, I feel like it's a pretty easy decision.
EDIT: The "neopronoun" question was added after I replied, or I missed it. I have never met a person who expected me to use them, nor have I ever encountered a workplace environment where policies required their use, so I haven't formed an opinion.
I've never met anybody who used neopronouns either, I've only heard about it online.
I wonder if there are any long term effects of forcing someone to say something that they consider to be untrue? Taken to its most hyperbolic extreme, it could be used as a form of psychological torture, like something out of 1984, where Winston is tortured for not accepting that four fingers being held up is five, or "Four Lights" from Star Trek.
To get one to renounce what they know to be true and accept whatever you say without question is probably the ultimate form of control and subjugation.
For emphasis: "taken to its most hyperbolic extreme".
edit: more realistically, you could say that transgender or gay people might feel like they are compelled to lie about who they are in order to fit in, or in certain circumstances. Surely, if we recognize this as psychologically damaging, then we should recognize all other types of forced lying to be similarly damaging.