In all likelihood, the stoppage of your dryer's drum had nothing to do with you drying your sneakers. A dryer can stop spinning at any time, and rarely does the problem coincide with drying specific items. Rather than focus on what was inside the dryer at the time the drum's spinning terminated, it's more productive to troubleshoot common spinning problems, as these are more likely to explain the problem.
A dryer's motor operates the belt to turn the drum. If the motor fails, the belt can no longer spin the drum, so it just stops. Contact a dryer repair technician to evaluate the motor's continuity with an ohm multimeter that measures the connectivity of electrical components. If low or no continuity is detected, he may recommend that you replace the motor rather than repair it; replacement is usually cheaper.
A thick, rubber belt encircles the drum at its center on most dryers. The belt spins the drum in a uniform circle with help from the motor, and supports the drum in the cabinet. Over time, the elasticity of the belt stretches until it finally reaches its breaking point. When this happens, the drum's rotation usually weakens, and you may hear a squealing noise from the belt as it tries to maintain its position. Your dryer's drum can't revolve without a belt, and you shouldn't operate your dryer if it can't spin. Swap out the defective belt for a new one.
A tensioner or idler pulley ensures that a belt can maintain a firm grip on the drum. Without it, the drum will either spin irregularly or stop spinning altogether. If the tensioner breaks, you may initially hear a grinding noise followed by knocking from the drum hitting against the cabinet until the drum eventually stops spinning. Review your dryer's manual for instructions on where to find the tensioner on your model. You'll likely need to remove the dryer's main access panel to view it. Once it's in sight, try to move the tensioner's wheel. It should move easily when you manually rotate it. If it doesn't, replace it. After it's changed out, your dryer's spinning ability should resume.
Broken Door Switch
It's possible that the weight of the sneakers tumbling in the drum could have broken the door switch. If the door switch breaks, the drum won't spin and the dryer will shut off. The door switch is a safety mechanism to keep you from putting your hand into a dryer while it's spinning and getting hurt. Open the dryer's main door. On the inside of the frame, locate the door switch. Press and release the switch manually with your hand. It should make a clicking sound when it's working properly. If you don't hear the noise or the switch appears damaged, replace it.