In all three examples, "this" should be replaced by "the". Also, the "$" sign is usually written before the number, even though it is said after the number.
With these corrections, option 2 becomes correct in both formal and informal English.
With these corrections, both option 1 and option 3 become correct in informal English. The contractions are consistent with informal English. In very formal English, the "who" should be replaced by "whom".
1b. He's the person who I owe $40 to.
1c. He's the person whom I owe $40 to.
2b. He's the person I owe $40 to.
3b. He's the person who I owe $40.
3c. He's the person whom I owe $40.
Another possibility is to put the "to" before "whom", instead of at the end of the sentence. With this word order, "to whom" is relatively common:
4c. He's the person to whom I owe $40.
To my (American) ear, the most natural options would be:
I owe $40 to him.
I owe him $40.