Using the TVM theory, at what interest rate would it make more sense to take out a loan versus paying with cash?

It makes more sense to take out a loan if expected returns on the invested money can be expected to be higher, with great confidence.

The "Time Value of Money" theory can be used to predict whether the current value of payments to be made in paying off the loan exceeds the received principal of the loan.

Using leverage to increase expected returns is a risky business; expectations can fail, as has happened recently.