Yes, bees will eat crystallized honey and there is no harm in feeding it to them. Remember that crystallized honey is not a modern invention. Bees have had to deal with it from the beginning and they know what to do.
In the depth of winter when bees cannot get outside, they use moisture that has accumulated in the hive to rehydrate the crystals. This moisture is the natural result of their respiration that condenses on cold surfaces within the hive. The bees take this water and “spit” on the crystals causing them to liquify. Bees eat candy boards, hard sugar cakes, and granulated sugar using the same method.
Some books claim that crystallized honey causes honey bee dysentery, but I do not believe those claims are correct. Honey bee dysentery, which is essentially bee diarrhea, is caused by honey having a high ash content. Ash is the stuff that remains after you burn away a sample of honey. You can think of ash as the “solids” that remain after you remove all the sugars. High ash content is associated with darker honeys.
Crystallization on the other hand is highly influenced by the ratio of glucose to fructose found in the honey. The higher the glucose, the more likely it is to crystallize. Other factors are involved as well, but this piece is critical.
Now, if you put these two facts together, you can see that honey with a high ash content that crystallized is more apt to cause honey bee dysentery than honey with a low ash content that crystallized. It would be easy for someone to feed high ash, crystallized honey to bees and conclude that the crystals caused the dysentery when, in fact, it was the ash that caused it.
If the crystallized honey you feed your bees is only part of their diet, or if the honey came from a variety of floral sources, it will cause them no problems. On the other hand, if you have many, many frames of crystallized honey with a high ash content, it could conceivably promote dysentery. It comes down to using some judgment about how much to give them. You can use the darkness of the honey for a rough estimate of ash content.
If you are trying to get your bees to clean up frames that contain crystals before winter sets in, put the super above an inner cover with a center opening. Uncap the crystallized honey, if necessary, and lightly spray the frames with warm water. The crystals at the surface will start to dissolve and the bees will be encouraged to move the honey down into the brood boxes, assuming they have room down there to store it.
If they still refuse to clean it up, it may mean they are still finding liquid feed—nectar, honey, or syrup—somewhere else. You can always move the crystallized frames down into the brood box or save them for later.