Subadditive functionals on the space of sample paths include suprema, integrals of paths, oscillation on sets and many others. In this paper we find an optimal condition which ensures that the distribution of a subadditive functional of sample paths of an infinitely divisible process belongs to the subexponential class of distributions. Further, we give exact tail behavior for the distributions of such functionals, thus improving many recent results obtained for particular forms of subadditive functionals and for particular infinitely divisible processes.