*Of which date's interest rate should I use?
This is ungrammatical. To begin the question with the preposition, you could say "Of which date should I use the interest rate?" This is, however, hopelessly stuffy sounding. Even if you want to sound stuffy, this is difficult to understand and should be avoided.
Which date's interest rate should I use?
This is a reasonable choice.
What date's interest rate should I use?
This isn't as good, but it wouldn't be surprising to hear a native speaker utter this question.
In some contexts, it might make sense to make the question longer. For example, you might ask Which date should I use for the interest rate? In other contexts, you might be able to make the question shorter, as in Which date should I use?