I'd like to tell you what do I think of the TV show.

小诀窍:what=the one that
可见上句可改为:I'd like to tell you (the one that) i think of the TV
show.那么该句 the one 代表想法(think of the TV的宾语补足语)that后引导定语从句.
在对what引导的从句进行语法分析的时候,虽然有时候用the one that不妥,但能更清楚的看出结构.
如上句:I'd like to tell you the one that do I think of the TV show
明显看出有毛病了多了以个 do
所以去掉 do 是正解.

I'd like to tell you what I think of the TV show.

去掉do