Some of the KidLitArtists were discussing the different ways they address this issue, and I've compiled their methods here, along with links to tutorials on other sites.
1. Scanning Trick
You scan your image twice, the second time flipping the image 180 degrees on the scanner bed. Then you use Photoshop's Auto Align feature to perfectly line up the two images (this feature is in CS3 and all later versions, it's found under Edit>Auto Align Layers). You can then adjust the opacity of the top layer until you have a nice blend between the two – and the opposite direction to the shadows means they sort of cancel each other out. You can also use layer masks to make more detailed adjustments to the layers.
|Image by Elizabeth Baddeley|
Good for small paintings that fit entirely on the scanner (unless you have the patience to scan your image in multiple pieces, merge them, and then do it again at 180 degrees!) and paintings that don't have a lot of fine lines and details that might not align perfectly.
2. Smart Blur Filter
For this technique, use the Smart Blur Filter (found under Filter>Blur>Smart Blur), and adjust the settings till you find the right ones to maintain a crisp image while blurring out the texture. (I personally like to duplicate the layer prior to using any filters, and then mixing the two using opacity settings and/or layer masks).
|By Gordon and Susan Pritchard|
Tutorial here: http://the-print-guide.blogspot.com/2009/02/removing-watercolor-paper-texture-in.html
I haven't used this one much myself, but I think it will probably work best when you have clearly defined edges, and a flat background.
The following two methods basically use different selection methods to isolate the background, and then either delete it, or (my preference) use a layer mask to hide it.
3. Channel Mask
This one removes the background entirely and makes use of the Channels Palette (open it with Window>Channels). You compare the separate channels till you find the one with the closest match to the information you want to select, then you duplicate that channel and use it to select that part of the image. Then Select>Inverse and delete the background. The tutorial goes through this technique in much more detail.
Similar tutorial at the bottom of this page: http://designshack.net/articles/graphics/a-complete-beginners-guide-to-masking-in-photoshop/
This one is good for when you want a white or transparent background, or if you want to replace the background with a different image. The tutorial includes a variety of other selection techniques.
4. Color Range
Andy Musser uses the Color Range feature in Photoshop to make a selection based on color, and then remove the background:.
Use Select>Color Range. Your cursor should be an eye dropper, click on the color you want to remove. To add more colors, or shades of the original color, select the “Eyedroper +” tool (in the Color Range controls) and click on them. The black and white image will show your selection, white is selected and black is unselected.
After creating the selection with Color Range, using the “Refine Edge” controls will refine the selection. Then invert the selection and use a layer mask to remove it.