When compared to shooting still photographs, does shooting video have any negative effect on DSLR sensors? For example, I wonder if shooting video may produce more heat, which might be harmful for sensors.
Shooting video, under normal conditions, will not have a negative effect on a DSLR sensor.
The sensor itself, being essentially a computer chip with no moving parts, can last a lot longer than the mechanical parts in the camera (shutter, mirror assembly, lens auto-focus motor, etc.).
A lot of (may be even most) cameras have over-heating protection and will shut down before the heat can cause any permanent damage.
Obviously if you use the cameras in extreme conditions that can damage the sensor (shooting directly into the sun for example) shooting video will expose the sensor to the dangerous element for much longer than when photographing.
All modern DSLRs have thermal cut-outs for the sensor which will shut them off if they start getting too hot. Repeatedly running the sensor near the thermal cut-out temperature will probably have some effect on the lifetime of the sensor, but that's more of a theoretical problem than a practical one - in a DSLR, it's generally the shutter that fails before the sensor.