Speculation: Could Kakashi's MS be incomplete Amaterasu? As is known, amaterasu is told to be fire from hell. Let's consider hell to be another dimension.
If amaterasu burns everything in its way, then it is possible, that it is completed by transfering it to another dimension, with "fire effect". What can support this speculation? When Itachi used it, he burnt everything, what was his eye focused on. When Kakashi used it against Deidara, he had to focus on him. Kakashi was able to use it against explosion. Itachi used it to counter fire jutsu. Uchiha's primary affinity is fire and primary affinity should be strongest and easiest to use. But Kakashi's primary affinity isn't fire. So we can speculate, that it is reason for no "fire effect".
Problem, of course, is right/left eye.