I have been filming lots of art work recently for a BBC strand and an independent production company. Although it is relatively straight forward to get this right, I thought I'd write a few notes up, as many people get this wrong and make a mess of it.
F-stop: It isn't a good idea to shoot paintings wide open as it is very easy for the edges of the paintings to be out of focus, while the centre is sharp (due to the relative distances from the lens). Stop the camera down a bit, 5.6 tends to be the sweet spot for most lenses, if you have enough light you could go further.
Framing: Set your tripod up so the head is right in the centre of the painting, from both a height and width perspective.
White balance: Light the painting and then take a reading from a white card right infront of the painting. It is essential to get this absolutely spot on. Experts will be used to the colours used in a painting, get it wrong and you'll know all about it. Usually there are no true whites on paintings, so the editor will have no reference point, meaning colour correction afterwards will be very tricky.
Lighting: Having filmed in lots of galleries over the past month I can safely say most gallery lighting is horrible. Typically painting are lit with harsh spot lights from the cealing giving very harsh hot spots. Get rid of these lights, turn them off if you can, point them away from the painting if you can't. Light with something very soft, I used a kino flo with a flozier over the top, which worked well.
Reflections: The hardest bit about filming paintings is dealing with reflections. Even without a glass frame, glossy paint will reflect a bit, so the light source needs to be a soft as possible. If the paintings are in a glass frame you could ask the gallery to remove it. If this isn't possible there are a couple of work arounds: 1)You can try filming the painting from a distance on a longer lens, this way you won't be able to see the camera in the relection as easily. 2) Another option is to film behind a black drape and just poke the lens through a gap.