I use a straightening iron quite a bit, the first thing I do with most ponies is use a rat tail comb to get rid of any knots and tangles, give it a wash in conditioner, shampoo if needed. Toy ponies aren't mammals, they don't sweat. Shampoo is for cleaning hair, conditioner adds things like oils, moisturizer, polymers to add a sheen or protect it, things like that, so you don't need to shampoo your pony unless it's been rescued from a multi year stay in a garage or somewhere else where it could have gotten dirty, but conditioner is a good idea.
After I condition the hair, I run a straightening iron through it while it's still damp, and on the lowest setting. After which I'll curl it, braid it, leave it straight, or do whatever else. I've yet to have any issues with the hair melting, but I've managed to give a lot of ponies straight hair.