Dying door panels, dash cap, etc.. is simple enough and common practice. Dying headlinners, carpet, seats, steering wheel, is not practicle. You'll wear the dye off the seat skins and steering wheel.
As I mentioned above I dyed a white bench seat black for my Barracuda. That seat was in the car for three years and close to 20,000 miles. The only place the dye rubbed off was on the piping on the drivers side where I slide my fat butt in an out. And as I mentioned it is an easy and simple job to touch up. It would still be in the car but it started to tear along the stiching so I got a new cover.
I didn't try my headliner because it was all torn when I got the car but it you can clean it it would be a good candidate for dying. I wouldn't try a carpet because you would spend close to what a carpet costs on dye. However, a steering wheel would work, people paint them all the time. A buddy of mine painted the steering wheel in his street rod with the same paint he painted the car and it has held up fine for years (orange over black). He used the same paint to paint the center inserts in the seats, it hasn't worn off but has cracked due to the flexing so I would say stay with the materials designed for dying.