I've taken a picture of a painting and about a quarter of it has a large amount of light glare on it as the painting is made with reflective oils. In Photoshop, is there any easy way to remove this?
If you don't have access to retake the photo using better lighting techniques, one method that might work with an existing photograph is frequency separation. It is often used when touching up portraits. Frequency separation is basically a way of separating the texture/details in the area of an image from the colors in the same area. You could use frequency separation to restore the color to the shiny areas without losing the texture of those areas.
For more on how to do frequency separation please refer to the following questions here at Photography on Stack Exchange:
How can I remove shine from skin caused by the light with GIMP
How do I take flattering photos of people with acne (scarring)?
How can I achieve consistent skin tones in my photographs?
Back in the film days the standard technique was called Cross Polarization, and it looks like the technique is still in use. The basic idea is to have two light sources at wide (but not too wide) angles from the artwork with polarization screens that are oriented the same way. Then on the camera lens would be a polarizing filter would be rotated until the reflections disappear.
You can find many illustrated examples of this technique searching for "cross polarization artwork photography."
(Note -- I've never done this myself, but it seems to be an established technique for reproducing artwork)