In theory there's nothing stopping you from using the whole 24 bits provided you scale the arithmetic correctly. You could even upset a few whiny purists and do the calculation in floating point if you like.
It might be worth working out the effect on temperature of a 1 LSB step in each case. If 1 LSB is still insignificant in the 16 bit result, then the bottom 8 bits really can be discarded.
This is a very odd part anyway, IMHO. I've never seen a temperature sensor whereby the host CPU is expected to perform this calculation; most report a value which converts directly into deg C with a simple linear scaling factor. TE isn't a semiconductor company, so maybe this is a part they've had designed under contract for some special purpose?
Its accuracy spec is very good, though, at least within a limited range. Maybe this is why they've had it designed? Do you need the accuracy, or might you be better off with a simpler to use part from, say, TI or Microchip?