What is the best way to paint a basement ceiling?
Can I paint the copper baseboard heating pipes? Do I have to use special paint?
Painting a ceiling is not that much different from painting any other surface. Take a look at the other questions in the painting tag here for lots of good tips. Definitely invest in an extension pole for your roller handle so that you're not holding your arm over your head for the entire time, and enough drop cloths to cover the ground in the area you're painting.
If you have bare copper heating pipes, you might want to think about wrapping them in insulation and painting that. I've got some of the regular pipe-wrap insulation in my house that the previous owner got paint on and it doesn't look any the worse for wear; I think it was regular latex paint, either gloss or semi-gloss. You can also get wraps that are specifically marked as paintable.
We've got steam heat, so the pipes get up north of 190F. In my basement, I ended up painting them with Rustoleum high-heat spray paint. Took a couple of coats, but they look good and made it through a cold northeast winter without any issues -- most notably, zero paint fumes when the heat is on.
I agree with @Niall that insulation is a much better option than paint. I just wanted them bare to add some warmth to the finished basement.
Painting the heat pipes with regular latex paint is not a problem or a fire hazard. I'd be sure the pipes were cool when you paint them just so the paint goes on smoothly and doesn't dry so fast it looks "clumpy".
I'm planning on doing the same thing for my basement and based on my research it looks like using a spray gun of some sort will save tons of time and aggravation.
Take a look at this too http://www.oneprojectcloser.com/painted-basement-ceiling-how-to-tips-and-lessons-learned/
If it's a flat finished ceiling, your typical roller will work.
If you're actually wanting to paint the underside of the floor above, the joists, and all exposed ductwork and plumbing, then Vitaliy has your answer...you want to use a spray gun to make quick work of it.