Examples of intracellular parasites of medical or economic importance?
As Armatus pointed out above, all viruses are obligately intracellular, and their medical and economic importance cannot be overstated.

Many bacterial species live intracellularly. The arthropod specific Wolbachia has a wide variety of consequences for its host, including alteration of reproduction and sex ratios, induction of reproductive isolation possibly driving speciation, and conferring protection from some viral infections.

Eukaryotes can be intracellular parasites as well: Malaria is a well-known example. Toxoplasma gondii is another intracellular parasite that uses cats as a primary host, but can use other warm-blooded animals (such as humans, rats, and birds) as intermediate hosts. T. gondii infection has been shown to induce behavioral changes in rats, causing an attraction to cats (thereby increasing the chance that the parasite will infect the cat and sexually reproduce). There is also some evidence that T. gondii infection in humans causes behavioral and psychological changes such as decreased reaction times, and links to depression and suicide.