Not if you install them properly and keep them clean.
Magnetic signs are intended to be used as temporary and/or removable mobile advertising and are manufactured to not cause any damage to your vehicle's paint job. The magnetic receptive side of our magnetic sign material is smooth, and when kept clean and applied properly, will not damage the underlying paint. However, with that being said, there are some guidelines to follow to ensure that your magnetic signs do not end up causing any damage.
First and foremost you need to properly install your magnetic signs by making sure the back of the magnetic sign is clean as well as the surface that it is being placed on. If you continually remove your signs and place them back onto a dirty vehicle eventually dirt and debris will build up on the surface of the magnetic sign which can cause scratches on your vehicle's paint. Magnetic signs and vehicles can be wiped down with a soft cloth and a mild detergent. Make sure to let the magnetic sign and vehicle surface completely dry before affixing your magnetic signs back onto the vehicle. And steer clear of placing your magnetic signs over rusty spots or flaky paint as this can reduce the magnetic signs receptive qualities. Also make sure to store your magnetic signs on a nice clean surface.
Secondly, you'll need to periodically remove your magnetic sign to avoid any sun damage. Since your magnetic sign is UV resistant continually leaving them on your vehicle could pose problems in the long run. Magnetic signs are meant to be removable, not permanent like vehicle graphics, and should be removed (and cleaned) weekly. When you place a magnetic sign on your vehicle, the underlying paint is covered up and protected from the sun, so if you leave the magnetic signs on for long enough the sun will fade the paint around the magnetic sign. This will leave you with an uneven paint job when the magnetic sign once was.
And be careful of what you buy when it comes to magnetic signs. Some other sign companies utilize thin and cheap magnetic material that has been known to permanently adhere to the vehicle surface. Make sure you are purchasing vehicle grade magnetic sign material. It may cost a little more up front, but be worth it in the end, because once a magnetic sign permanent bonds with your car surface it's very hard to remove the magnetic sign without completely damaging the paint.