If a light fixture gives off a burning smell, disconnect the fixture until you have determined the source of the odor.
Overheating electrical wires and devices often emit a burning smell. Don't use the fixture again until a professional has inspected and repaired it. A fluorescent fixture may have a ballast that has failed and is spilling tar. For typical incandescent light fixtures, the burning smell may occur if you're using an oversized bulb. Check the rating of the fixture and the wattage of the bulb. The rating will be inside the fixture, near the bulb. Never exceed the wattage recommended.
You might also have a loose electrical connection at the splice or in the outlet box, or a loose screw or lamp base. A loose connection can create excessive resistance to electrical flow, and the resistance causes heat. Excessive heat makes metal connections expand and contract, loosening them further. This heat can damage insulation and even start a fire. Sometimes, when such excessive heat melts plastic, the problem area emits a misleading "dead animal" smell.
If you notice any strong smells near outlets, electrical boxes, or light fixtures, they may be due to an electrical problem. Call an electrician to evaluate and fix the problem. In the meantime, do not use electrical power in that area.