|[Note: I am neither a doctor nor a certified personal trainer. Those with health concerns should consult a physician.]
Simple answer: No.
Running is a great workout. Lungs, heart, nervous system, etc. get jumping. It need not be totally eliminated as age sets in, but there are concerns to be aware of.
The first concern is that you could pull or tear a ligament or muscle. As humans age, the body tends to get drier, harder, less elastic, etc.
The solution to this problem is to be sure that you adequately warm-up and cool down. As you age, increase the amount of time that you spend stretching, walking, jogging, etc. before and after. If you feel any pain in the body, do not push yourself. There is a difference between pushing yourself when tired and pushing your hamstring to the point of tearing. Warming up gets the blood flowing throughout the body, getting you lubricated, and minimizing the risk of injury. This goes for all ages, all exercises, but the need increases as you age.
The other concern with running is the pressure put on your knees, hips, lower back, etc. as you run. Animals that are big runners (bolting to avoid predators) have hooves, less surface area to connect with the ground. Animals that climb trees to avoid predators have big feet. Humans are the latter. In evolutionary time, it wasn't too long ago that primate humans became bipeds. Marathon runners often have joint trouble over the years. I would not eliminate running altogether, but I would not regularly run over an hour a day every week either. Incorporate cardiovascular exercise that does not put pressure on the the same joints, such as swimming, climbing, etc.
also, terrain- I would favor running on grass or dirt trails over hard blacktop. A little less impact on the body.
shoes- also should lessen the impact. Make sure the soles are very spongy to soak up the shock. Shoes should also be neutral, not turning the feet one way or another with weird supports.
It might sound weird, but some styles of yoga, such as bikram or ashtanga are very tough workouts that also involve lots of stretching. Kareem Abdul Jabar and other sports figures did/do rigorous yoga as a means of preventing injuries. Also, drink lots of water and eat lots of raw fruits and vegetables to stay hydrated.
Summation- In most cases, running need not be eliminated as one gets older. Just make sure you do not overdo it. Listen to your body, watch for pains. Finally, make sure you adequately warm-up and cool down.