Can plants break down cellulose for energy?
Plants do not produce extra cellulase to break down cellulose for an energy boost when they are grown in dark conditions (that I know of). But remember, cell walls are the structures maintaining turgor pressure in the cells, so breaking them down would be very costly to plant survival. A couple notes: plant cell walls have more components than just cellulose, and they ARE dynamic. In dark-grown conditions, plants elongate (etiolation) during which time, enzymes (expansions) break down cell wall bonds to allow the cells to elongate the stems of the plants. Elongation is an evolutionary advantage to find light when things get shady. 

To read more about what the cell wall is made of, and the function it provides to the cell, try watching the Khan Acadamy video about cell walls. Light signaling is an entire field in Plant Biology (example). Read more, if you are interested in how immobile plants handle shading from other plants or periods of darkness. Plants do break down proteins for reuse! Read more about plant vacuoles if you're interested in how they recycle and break down some molecules and proteins when needed (related to your question's parallel to animal cells; although, not cellulose breakdown to replace photosynthetic sugar output).