How can I speed up the rate at which putty dries?
Heat will make the putty softer, which is why you can heat it up to remove it.  Putty pretty much stays soft for years.  Initially, it doesn't really dry out so much as oxidize which forms a skin on it. 

If you absolutely must paint it right away, you'll probably has to use one of the latex based glazing compounds. If you want to use a traditional one, paint it as soon as it forms enough of a skin that the paint will stick to it.  Also, I think you probably should be using oil based paints, but check the instructions on the putty.

If you let use know specifically which putty you are using, that might help get better answers.