You have two issues going on with these dyed floors. The first is moisture-vapor transmission, or MVT. This is the cause of the white crystal growth and white powder residue you are seeing. On indoor surfaces, it usually occurs along the outside walls, or near joints and cracks where moisture can easily penetrate and hold. Did the floors go through a wet winter or spring? Are the areas where the white residue is occurring near walls or exterior landscaping that is wet or holds water, such as lots of dirt built up alongside the outside wall? Water in concrete, whether in liquid or vapor form, picks up excess salt and carries it to the surface. This is known as efflorescence. (For more information, see Efflorescence on Black Concrete.) The pressure from the moisture vapor and salts either doesn't let the dye penetrate to start with or actually forces it out of the concrete over time. It looks like the latter is occurring with your slab.
The second issue is "tape pull," caused by the plasticizers in the glue on the painter's tape penetrating into the sealer and creating a very strong chemical weld (see Tape Marks on Concrete Floors). This can occur very quickly, and even with weak painter's tape can form a very strong bond. Time, heat and pressure make this worse, as well as the type of tape that is used. The bond of the dye to the concrete and the sealer to the dye and concrete also play a part. Usually we find that when blue painter's tape pulls up the dye and sealer, as in your case, there were underlying issues with the original dye penetration and adhesion. This also could be related to the MVT issue discussed above. It is best to never put tape on sealed concrete.
Now, here are some ways to fix your problems. If the dye failure is isolated to a few small areas, you may be able to clean the floor with a mild acid, neutralize the area, let it dry, and then touch it up with tints, dyes or a tinted sealer. The problem with this approach is that it does not address the root of the problem, which is MVT. If the source of the moisture is something you can control or fix, such as a broken pipe or sprinkler line or regrading of the exterior landscaping, then taking care of those issues should stop the problem. If you can't stop the moisture, the problem will probably occur again, which means the floor will need to be treated and waterproofed. This usually requires stripping back down to bare concrete, applying a waterproofing sealer or membrane, and then coming back with a topping that can be dyed or sealed. The repair route take you take will have a lot to do with how widespread the issue is, where the moisture is coming from and how much time and money you want to spend to fix the problem.
To cover the marks from tape pull, try faux painting the spots with tints or dyes in colors that will blend with the surrounding areas.
Glue from painter's tape bonds with the sealer
and pulls off the sealer and dye.
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