There are things one can do to help with white backgrounds, but I've not found one that is useful in all
Begin with a good scan or photo, one where the white is close to the highest value.
Then when processing in Photoshop, apply a Curves layer and slide the white down to, or slightly INTO, the spike in the histogram on the white side. If this changes the contrast in the main image, use the curves line to change that. It helps to put a couple points up near the whitest end to prevent the white from being reduced in value when making changes elsewhere on the curve. This will work unless there are pale colors.
Another method you should experiment with is the Select > Color Range tool. You can use this two ways. One is to select the white background (the background is never completely white so you'll have to use the +dropper), then either delete that or fill with white. I'm continually amazed at the times when I've hit the delete key and it gives me a pale colored background. Has to do with the droppers on the left, the blend mode, and the opacity, etc., so recently I've just been going to File > Clear. Nothing else to check :). The other way is to select the color in the image, then reverse the selection and clear. Often, however, no matter how you do the selection with the Color Range, it will include part of the image. If this happens, and it's only a certain part, just add to or delete from the main selection with one of the selection tools. I usually use the Polygonal Lasso to do this. It sometimes helps to narrow the "Fuzziness" slider on the Color Range tool, which selects a narrow range of colors.
When all else fails, take the eraser tool and carefully erase around the outside of the painting/image. Change point size when needed and make sure the eraser is at 100%. (Duh, been there, done that. Once.) Just erase out to about 50 pixels or so. Start at one spot and move around the outside of the image in one direction. If there's a smallish closed-in area, just erase out to where the white background is more open. When you've finished this, use the Polygonal Lasso tool, or similar, to make large selections out to the edge, and within the 50mpix erasures. Delete back to background color, or go File>Clear. Do this on a large scan of the piece, as any small errors in erasure will be minimized when downsized for the web. If this is for a client, one just needs to be very careful in the initial erasing. A Wacom tablet helps a lot. This is tedious, but it works. It won't take as long as you might suspect. Others would use the pen tool or one of the magnetic tools to do a selection, but I find the eraser easier to correct, and much easier to stop using for a few minutes to refresh my coffee :).