Origins
Relief treats suffering as something heavy. The word comes from Old French relief, from relever — 'to raise again, to lift up' — from Latin relevāre, composed of re- ('again') and levāre ('to raise, to lighten'). To relieve someone is to lift a burden off them. To feel relief is to feel that weight rise.
The Latin levis meant 'light in weight', and it produced a family of English words built on the same metaphor. Levity is lightness of mood. Elevate is to raise up. Alleviate is to make lighter. A lever lifts heavy objects. All involve the upward movement of something that gravity pulls down.
The sculptural meaning — bas-relief, high relief — uses the same root differently. A relief carving is one where figures are raised from a flat background, literally lifted out of stone. Italian rilievo, French relief, and English relief all describe this raising.
Latin Roots
Relevant belongs to the family too, though the connection is less obvious. Latin relevāre also meant 'to bear upon', and something relevant is something that bears on (lifts up) the point under discussion.
The word entered English through Anglo-French in the 14th century, initially as a feudal term — relief was a payment made by an heir to a lord. The emotional and humanitarian senses developed alongside, always carrying the image of weight removed.