how many days do you think they will need in shanghai?（这句对.）
how many days do you think will they need in shanghai?
原因：
do you think/believe/suppose常在疑问句用充当插入语.因为一个疑问句只有一个是疑问结构（也就是do you think,）而不能再使用第二个疑问结构,所以,就用they will 了.
又如：
What do you think they are doing?
如果去掉do you think,改成：
What are they doing?