Obviously, to do an interpolation search, you need some type of key for which more than ordering is known -- you have to be able to do computations on the keys to estimate a likely distance, not just compare keys to determine which is greater or lesser.
As far as properties of the dataset go, it mostly comes to one property: a likelihood that the keys are reasonably evenly (or at least predictably) distributed throughout the range of possibilities. Without that, an interpolation search can actually be slower than a binary search.
For example, consider a data set with strings of lower-case letters as keys. Let's assume you have a key that starts with "x". An interpolation search will clearly indicate that you should start searching very close to the end of the set. If, however, most of your keys actually start with 'z', and almost none with anything from 'a' though 'y', the one you're searching for may actually be very close to the beginning of the set instead. It can/may take a considerable number of iterations before the search gets close to the beginning where the string starting with 'w' reside. Each iteration would remove only ~10% of the data set from consideration, so it would take several iterations before it got close to the beginning where the keys starting with 'w' actually reside.
By contrast, a binary search would start at the middle, get to the one-quarter mark at the second iteration, one-eighth mark on the third, and so on. Its performance would be nearly unaffected by the skew in the keys. Each iterations would remove half the data set from consideration, just as if the keys were evenly distributed.
I hasten to add, however, that it really does take quite a skewed distribution to make an interpolation search noticeably worse than a binary search. It can, for example, perform quite well even in the presence of a fair amount of localized clustering.
I should also mention that an interpolation search does not necessarily need to use linear interpolation. For example, if your keys are known to follow some non-linear distribution (e.g., a bell-curve), it becomes fairly easy to take that into account in the interpolation function to get results little different from having an even distribution.