Under normal use, no—recording video is not considered harmful to a DSLR sensor. The sensor is an electronic chip with no moving parts, and in practice it usually outlasts more failure-prone mechanical parts such as the shutter or mirror system.

Video does keep the sensor active for longer, so it can generate more heat than shooting stills. However, modern DSLRs generally include thermal protection that will shut the camera down if the sensor gets too hot, preventing permanent damage.

In theory, repeatedly operating right near the thermal shutoff limit could reduce sensor life somewhat, but this is more of a theoretical concern than a practical one for most users.

The main exception is extreme conditions—for example, pointing the camera at something that can damage the sensor, such as the sun, for an extended period. Video would expose the sensor to that hazard for longer than a single still image.

So for everyday shooting, video recording should not negatively affect your DSLR sensor.