The brick color is hideous. Additionally there are some missing bricks that will be replaced first and would be hard to match.
Is there a particular type of paint that should be used on brick?
I did this exact thing a few years ago in a friend's house. The steps I took:
Remove lime efflorescence from the brick by spraying on some diluted white vinegar, then scrubbing with a stiff-bristled nylon brush. Leave to dry.
Prime and seal (there were some smoke stains on the brick) the fireplace with Kilz latex spray primer. Leave to dry.
Paint with two coats of a pure white exterior gloss latex paint that I had left over from painting some wood trim. The rough texture of the brick meant that the final surface wasn't super-shiny.
This is how it turned out (sorry about the photo quality):
Most paints should work with a decent primer but you can also get masonry paints. They're typically for outdoor use, however, and are designed to help the concrete breathe, which is less of an issue on an interior fireplace, so may be overkill.