John asked Jane to help him move the dining table, but she refused to

(E) in my opinion. Took me 54 secs.
Here, it is a pronoun, it can only refer to a noun. But here it is incorrectly trying to modify/refer to an action. It's better to use "to do so/doing so". Thus, reject option A.
That again is a pronoun and it refers to a previously mentioned noun. And thus can not refer back to a previous action in the sentence (here the action of helping John in moving the dining table). Hence, reject option B too.
This corrects the pronoun ambiguity clearly but introduces a slick error. If you didn't pay attention and were in a hurry, you would mark this answer as correct, ignoring the subtle change of table to tables. It changes the original meaning where John needs help with moving the dining table, not tables. Hence, reject this cheeky option C.
This option has errors from all the options above. Easiest to reject.
This is the remaining option and has the least number of errors in it too. This option correctly uses to do so to refer back to a verb and does not make any blatant pronoun errors like in Option A and B. Hence, this is the right option.