People that need to keep accurate time keeping, I suppose.
I expect that for a whole host of applications having a second that is randomly different from all other seconds would be irritating. Anything dealing with automation on a assembly line would probably be irritated to know that their devices are going to have to deal with time units that are shifting goal posts. Avionics was mentioned above as well as a couple other things that I won't bother repeating.
The whole point of the leap second is to keep a second a second. When a second is not a second then what do you do to deal with that?