option C

A.it can be potentially dangerous for people with tight budgets, whose --- Incorrect use of pronoun "It" with plural subject "variable rate mortgages"..
B. they can be potentially dangerous for people with tight budgets, because their --- Presence of" their" is ambiguous
C. they are potentially dangerous for people with tight budgets, whose --- correct { "whose" is correctly pointing to "people with tight budgets"
D for people with tight budgets, it can be dangerous, because their --- same as A
E. for people with tight budgets, they can be potentially dangerous in that their -- Incorrect use of that

Correct answer must be (C) for the highlighted errors in other options....

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Is 'whose' in option C not referring to 'budgets' ?

Good question
Answer is C
A it in is wrong as it refers to a plural noun "variable rate mortgages"
B their is ambiguous as it can refer to both "variable rate mortgages " and people.
C Correct as whose is used to refer to people .
D it is wrong
E their is wrong

In choice (B), isn't "can be potentially" redundant?

Yes, the redundant "can potentially" is a problem in A, B, and E.
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A.it can be potentially dangerous for people with tight budgets, whose - "Variable interest rates" is plural we cant use "it" - Eliminate A
B. they can be potentially dangerous for people with tight budgets, because their - "can be potentially" is redundant. Use either can be or potentially not both - Eliminate B
C. they are potentially dangerous for people with tight budgets, whose - Keep C - Best available choice
D. for people with tight budgets it can be dangerous, because their - same as A - Eliminate B
E. for people with tight budgets, they can be potentially dangerous in that their - pronoun ambiguity with "they" as it can refer to "budgets" or "Variable interest rates" - Eliminate E

Looking forward to receiving your feedback on my explanation.