How do you cook a hard cheese to dry it out without melting it in a home kitchen? Which cheese is suited to this purpose?

I don't see how you can make cheese crispy without first melting. You can, however, easily make a cheese tuile from a hard cheese, like parmesan. The cheese is grated, then baked, where it melts. It is removed from the oven and crisps as it cools.

Here is an example with a description of the process, and a picture of the result. There are plenty of others, just google "cheese tuile."

you would need to use a grill or a broiler to crisp up the cheese (look up "gratin" )

Some cheese are better at it than other.

Parmesan or cheddar can be use to make parmesan or cheddar chips.

be careful, there's a fine line between crunchy and burnt.

This looks to me like 2 (and a bit) questions: one on drying, and one on making it crispy. But they do overlap.

You can apparently - I've never tried it - dehydrate cheese in a food dehydrator; I don't know what texture you'd get, but I would guess not crunchy. I have bought dried grated pseudo-parmesan (good for camping). Because it's grated it's hard to tell the texture, but I doubt bigger pieces would go crunchy This rather long post does imply crunchiness is possible.

In summary they suggest drying it cool and slowly. If you're starting with a hard cheese made from pasteurised milk, it can be stored at room temperature anyway - max 30°C, which is often the lowest temperature on a food dehydrator. Much warmer and it goes oily (cheddar, in my experience).