There seems to be a mistake in the quiz of lesson 31 in the question:

"Translate the following sentence: "You mustn't lie!"

a.　むだづかいしちゃいけませんよ。
b.　むだづかいしちゃだめだよ。
c.　むだづかいしちゃだめ。
d.　むだづかいしてはいけません。
e.　The four sentences are incorrect. (The app says this option is the right one)

Are these four sentences not all correct? Or is there some glaring mistake I'm not seeing? いけません and だめ both take the -て form right?

It seems you're mistaken about exercise.
To: "You must not lie!"
The answers are:

1 - うそをついちゃてはいけませんよ。
2 - うそをつくちゃいけませんよ。
3 - うそをつくてはいけませんよ。
4 - うそをついてはいけませんよ。
5 - The four sentences are incorrect

The answers you have are for another exercise:
Which of these phrases is incorrect?

1 - むだづかいしちゃいけませんよ。
2 - むだづかいしちゃだめだよ。
3 - むだづかいしちゃだめ。
4 - むだづかいしてはいけませんよ。
5 - The four sentences are incorrect