If by “the camera can take a certain number of photos”, you are referring to the expected lifespan of the shutter. No, I don’t think so. The “shutter” during video is electronic, offloading the image. Not physically opening and closing the shutter.
There is of course potential for increased wear on the electronics and circuitry. If your camera gets really hot while recording I would limits its use for video. But IMO it is rediculous to worry about whether or not using a cameras standard functions, in normal conditions, are wearing it out. IT’S A CAMERA! Use it up, make your investment in glass, by the time the camera wears out it will be long since antiquated and time for a newer better body anyway. Unless your talking about repeatedly recording for extended periods of time in extreme conditions.
Just my thoughts, not like I’m an expert on DSLRs lifespan. Don’t even own a DSLR but I’ve been shooting video for the majority of my life.