I can't tell you for a human, but I can tell you for a deer (mammal of roughly similar size, although I would guess more resilient and tougher than a person in this regard).
Big part of it is going to depend on what you get shot with. An arrow that just slices the outside of the lung? A pistol? A relatively low-powered rifle like an AR-15 shooting a fmj round? A hunting rifle like a .30-06 shooting a bonded partition? It makes a big difference. The faster (and, to a degree, the more kinetic energy) imparted by the projectile, the larger the secondary wound channel and the more hydrostatic shock the hit causes. Short version is that a rifle round can impair and damage tissue for a meaningful radius outside of the actual hole the bullet drills.
In general, a deer hit in the lungs well with a bullet will lay down within 10 seconds and die within a minute or two. With a marginal hit that scrapes the lungs, depending on what else is hit, the deer might stay on its feet for a few minutes and stay alive for several hours. Thing about the lungs (at least in deer anatomy) is that there are so many large blood vessels and other important organs packed around them (heart, spine, liver) that it's really tough to be close to the lungs without also putting a hole in something else important. That's one of the reasons lungs are a primary target for hunters.
Hope that helps.