No harm will be done to your dishwasher if you use fairy liquid dish soap in the dishwasher instead of regular dishwasher detergent.
The only difference in regular dish detergent and dishwasher detergent is that the dishwasher detergent is made to create less suds than regular dish detergent.
The less suds means that the dishes will be washed better in the dishwasher and all the detergent will be rinsed away easily with the dishwasher.
If you use regular dish detergent in your dishwasher such as fairy liquid dish detergent instead of dishwasher detergent then the soap may not get removed fully from the dishes.
So you may need to rinse the dishes again to remove the excess dish detergent leftover on the dishes.