Copper is not abrasive at all. In fact, in stronger alloys such as bronze it makes a bearing sleeve for low wear, direct metal-to-metal contact. Copper is not very strong and cutters go through it easily. However, like other soft metals (such as aluminum), there is a chance you can jam up the cutting teeth or a grinding stone particularly when high speeds are used. It is generally not a problem with cutting disks.
BTW, the Dremel was good in its day, and it's still useful in itself, but it's now totally obsolete and overpriced. Black and Decker RTX, the one with the continuously variable speed, is WAY better. AND cheaper. It's got a regulated speed, better form, maybe 50%-100% more power at full speed, and at the low speed settings, around 5x the power.
Having the speed regulation and high torque at low power is extremely important to the versatility of the tool. Most of the pitfalls with rotary tools come from using them at too high a speed for the job. Using brushes that fly apart if used at too high a speed, avoiding melting plastics, avoiding burning fiberglass, avoid shattering the cutting disks- gotta have a regulated speed and the torque to back it up.