For your original question, interpolation does not increase the available precision, so you only need to worry about rounding error. If your input is float32 or worse, float32 is fine for the interpolation output.
Regarding your update about combination of values in reconstruction: the ideal conditions for a combination of independent measurements is repeatedly measuring the same quantity, in which case you can gain log2(N) bits of precision - for 32 independent measurements of the same quantity at 16 bits each, you can gain 5 bits, leading to a total of 21 bits. float32 has 24 bits of precision, so it has room to spare.
However, if they are independent measurements but there are some weighting factors involved, it will not approach this ideal. If they are not of the same quantity, such that there is cancellation involved, you can instead have less precision than the inputs. This also ignores the more practical issue of SNR.
As a more intuitive guideline for this, float32 has a precision ratio (take a value and the closest next representable number, divide the difference between them by their average) in the range [2^-24, 2^-23] = [10^-7.22, 10^-6.92]. For comparison, 7 significant figures in scientific notation has a precision ratio in the range [10^-7, 10^-6] (because 1.000002 and 9.999998 both have the same absolute step to the next number, but differ by an order of magnitude).
So you can just ask yourself: does it make sense to have more than 7 significant figures? If not, store it as float32.