取消 = cancel events that are abstract / formal / technical / neutral in nature - 取消制度,取消预约
消除 = clear and eliminate negative effects, in order to protect the integrity of something else, implying an act of objective justice 消除不良影响、消除异味
清除 = actively delete / eliminate something negative, with an emphasis on subjective intention 清除威胁、清除障碍
撤销 = retract some (neutral) effort to return the original object a former state