Let me preface this by saying: yes, you need rest to optimize recovery and performance in any sport you're participating in. This includes lifting, running, gymnastics, whatever it is. How much rest you require depends on the sport and the demands.
For example, strength training places a very high demand on musculature, connective tissue, ligaments, tendons, and bones. These do not repair within 24h. To be honest, research shows that you continue to have increased biomarkers for muscle damage over a week after a heavy lifting session. In addition, you have to elicit a full blown immune response to heal well. That takes more than a day to get going. It's a particularly important process that results in absorption and breakdown of insufficiently strong connective tissue, adds sarcomeres in parallel to your muscle fibers, and builds you up stronger. While the process isn't complete when you have one day of rest, it is sufficient such that the body has made some headway.
For something like running, the demands are very different. Considering it's largely an endurance sport, the demand on a well-trained runner's muscles are fairly low compared to strength training. However, you have to consider that the demands of running are very different. Most importantly, you are utilizing your cardiovascular system much more intensely. The impact forces are quick and stress the bone and joints with more overall volume than strength training. The forces, overall, are lower however. In this sport, the goal is to avoid excessive volume so that you can appropriately improve your bone density and cartilagenous thickness and resilience.
It all depends on what your level is (untrained, beginner, intermediate, well-trained), your recovery potential (genetics), your sleep, your nutrition, etc. If you're going to be lifting heavy most days of the week, it better be different muscle groups and varied - or it must be low volume and/or low weight. If you don't, you run the risk of stressing an insufficiently healed tissue. That's where you get muscle strains, ligament sprains, joint pain, CNS fatigue, etc.