Yes, they absolutely do. This being said, common sense can keep them from being burnt in too much.
Remember to calibrate the panel when you get it. This is the single most important thing. Most of the time, they come from the factory with brightness and contrast cranked up too high.
Also, burn in relates to overall panel life, so today's panels are far FAR more burn in resistant than those made even a year ago.
The earlier plasmas had a 10-20k lifespan. Those burnt in very very easily. Today, they have a lifespan of upwards of 60k hours. Next gen will be even longer and harder to burn in.