Do concrete deicers damage concrete? The answer is no. It's the misuse of deicers that causes damage to concrete.
The two most common types of deicers are sodim chloride (rock salt) and calcium chloride. They do nothing when dry, but form a brine when combined with miosture. This brine penetrates down to the concrete and spreads out, melting and breaking the bond between the ice and concrete. There is also heat created when the salts dissolve (exothermic reaction) which facilitates melting.
The common misconception in using these deicers is that you should wait for all the snow and ice to melt. But the proper way is just get the bond of ice and concrete broken, then mechanically remove the rest of the snow and ice with a shovel or plow.
So why not just wait until it all melts? Concrete's biggest enemy is freeze and thaw cycles. Water expands as it freezes. This expansion pressure taking place in the pores of concrete can bust it apart causing spalling. Leaving salt sitting on concrete increases the number of freeze and thaw cycles thus increasing the chance that the concrete will be damaged.
On the package of most concrete deicers, the instructions will tell you to mechanically remove the ice and snow once it is broke loose from the concrete. Not only does that prevent damage, but keeps people from tracking in that unsightly salt residue.