My experience of heating coffee in a microwave is very negative, although I cannot explain the mechanism for it turning out so awful. It may be the effect of the hot coffee cooling down, rather than the effect of the reheating.
We’ve all experienced coffee that’s been sat on the hotplate for too long; bitter and acrid. You risk this “burning” effect if you heat cold coffee on a hotplate.
Cold press coffee is generally brewed very strong indeed; about twice as strong as espresso. Hence you can add boiling water at a ratio of, say, 3:1 and have an acceptably warm cup of coffee – and this is what people usually do.
If you want to microwave for extra heat, just microwave the full-strength cold-brew for 10 seconds or so, before adding water from the kettle.