A properly functioning dryer should never smell like burnt rubber or elicit a burning smell of any kind. Stop using the dryer when an unpleasant odor is detected until you can determine its cause. If you're unable to troubleshoot the issue on your own, contact a dryer repair technician who can inspect the dryer and address the problem.
Closely follow care instructions on clothing labels when washing and drying them. Certain synthetic fabrics, like Lycra, can't tolerate high heat as well as cotton can, and drying them on a high heat setting can scorch clothing. If clothing becomes scorched, it can produce a strong smell that's similar to burnt rubber. Lower the temperature setting on your dryer when drying multiple items of various fabrics to avoiding burning them.
Drum Fails to Spin
A dryer drum is designed to spin as hot air is dispersed into the dryer. The rotation enables clothes to move continuously so that hot air can dry items uniformly and clothing doesn't idle on a hot surface. If clothes remain stagnant, extremely hot air can burn clothes in certain areas and create a burning smell. Verify that a fuse hasn't blown or that the safety device that prevents a dryer from overheating hasn't triggered. Both kinds of component failures can cause a dryer to work partially but not fully until the parts are replaced.
Inspect the dryer's main power cord for defects. A frayed or exposed wire from the power cord can produce a burnt rubber smell when the dryer is on. When electricity is transmitted through a cord with a faulty wire, it can melt the rubber protective clothing that surrounds it. Replace a bad power cord with a new one. Discontinue use of your dryer until the power cord is replaced to prevent electrical shock.
A collection of lint in the dryer exhaust system can cause small fires to develop inside the dryer when it's on. Heat from the dryer gets hot enough to ignite tiny fabric pieces trapped in the exhaust system. As soon as you notice a burning smell coming from your dryer, stop the dryer mid-cycle and disconnect power to the appliance. If the fire is visible, use a fire extinguisher to put out the flames. In most cases, small fires in the dryer or exhaust system will burn out on their own after the lint burns completely. Consult your dryer repair manual for instructions on how to thoroughly clean out the dryer's exhaust system.