Why do tick bites occasionally cause the bite area to become inflamed and itchy long after removing the tick?

Allergic: Reaction to the bite. Also could be a low grade skin infection as well.

Irritants in saliva: As the tick feeds to obtain a blood meal, some of its saliva is injected into the human or animal it is attached to. Components of the saliva can be irritants and cause an inflammatory reaction, even after removal of the tick. Tick bites, like other puncture wounds can occasionally become infected also. And, in areas of deer ticks, the bite area has to monitored for the lyme disease rash.