Just like anything: consistent and repeated hard usage of a mechanical object will shorten its lifespan. How much? That depends.
So the best, short answer to Kevin's, the original poster's, question is a definite "Yes."
The long answer is more complicated.
Bicycle frames were and are never designed for, their technical specifications do not include, nor is any testing done to determine their suitability for or capacity to withstand repeated usage on a stationary trainer. Period.
Testing for use in a stationary trainer would not be difficult for major frame manufacturers, but it would change the design specifications and make bike frames harder to manufacture and even more expensive, and most bikes are ridden outside. So they don't do it.
I have personally seen metal frames fatigue prematurely, and all types of frames of different material types fail (i.e. break) from excessive use on a trainer. I've worked for a major manufacturer, a regional distributor and several different retail shops, so I've seen a lot of wrecked frames for a variety of reasons.
The lawn dart catcher is right on about rollers but there is no set of rollers made that is capable of providing the level of sustained high resistance to allow the development of high power required by most competitive cyclists. They are used by professionals for base building (endurance) workouts and to develop superior bike handling skills.
Damage resulting from sweat can and will occur if not wiped off, so that was also a useful response.
Other posters who talk about skewer damage are not answering Kevin's question. The skewer is not part of the frame.
Finally, the extent of frame damage will depend upon the rider's size and strength and their actual pedaling efficiency. Remaining seated as Moz recommends is really not a factor because it depends upon riding style. Many testing facilities have observed that talented and/or experienced riders who climb well produce lower lateral stress on the frame when out of the saddle. These riders inherently know or have learned how to apply torque where it really matters: on the pedals.