Learn English – “Heat up” or “heat”

Various grammatical sects profess various dogmata about phrasal verbs, but have to doesn't really fit any of them. That to isn't an ordinary preposition or adverb married to HAVE—it's a some-other-sort-of-particle married to the infinitive which follows.

One simple test:

^ok I can’t put up with him. → It’s him I can’t put up with. → He’s who I can’t put up with.

^ok She’s into metal. → It’s metal’s she’s into. → Metal’s what she’s into.

 ^∗ We have to hurry. → It’s hurry we have to. → Hurry’s what we have to.

The same thing is true of the other periphrastic modals: BE able to, BE about to, BE going to, and so forth. They don’t behave like the other things that are called phrasal verbs so you need to call them something else.

My answer will be for U.S. English.

Let's say I'm giving someone instructions:

I want you to write out a list of possible solutions to your problem.

This means I want you to take some ideas that are in your head, and go to the trouble of putting them down on paper. Implicit in this sentence is the assumption that if I am not quite explicit, the person will probably not bother writing anything down. I am imagining saying this to one of my children.

I can't imagine saying this in the context of creating a CV. For a CV, I suppose I might say

Please write up a draft of your CV and I'll help you format it.

The copying meaning mentioned in the comments sounds reasonable too, but in this day and age that doesn't seem very common to me.