It would be a mistake to dance to their tune.和他们周旋将是个错误。

名词tune，曲调，调子。整个习语dance to one's tune，直译就是“跟着某人的曲调跳舞”，用中文说，就可以是“听某人指挥，跟某人周旋，被某人牵着鼻子走，对某人唯命是从”之类。