This could definitely cause a dryer no heat problem and a potential house fire. Purchase a new dryer cord and replace it. The following link explains how:
Follow the 3-prong wire color code provided. If the new dryer cord does not have any colors (all gray) simply make sure the center conductor gets connected to neutral (WHITE) at the terminal block on the back of the dryer. There should also be a grounding strap or ground wire (GREEN) attached to the neutral at the temrinal block. Each outer wire will then get connected to either of the two hot (RED and BLACK) terminals.
The reason your dryer still may tumble in spite of a bad dryer cord is because the drive motor only uses 110-120VAC while the heating circuits require the full 220 service. If one of the dryer cord conducters is burned in half, you may only be getting half your input voltage to the dryer. A voltage reading across the two hot terminals (left and right slots) at the wall receptacle should read 220-240VAC. You should also read this same voltage across the red and black wires at the terminal block. If not, you have a bad cord.
If you have any questions, please let me know. I hope this helps you.