I have a laptop that has a 1X DVD-R drive, and I'm wondering if I use 8X DVD-R discs, if they will harm my drive.
Nah, as far as I know, the speed of the disc is the maximum speed the disc should be written at, anything slower is fine, just problems could be caused if writing faster than the disc is meant for, could cause errors in burning and writing could be inacurate. Slower writing is normally much better and reliable anyway.
The only problem you might encounter is with Pioneer burners. I recall that a 103 model needed a firmware upgrade or using 8X media would destroy it. I would want to make sure my firmware was up to date; if Pioneer had this problem other might have too. Also you drive could have difficulty recognizing newe media that was produced after the firmware.