Like many responses in the fitness space, the answer is yes... and no.
I say this because working out and fitness, in general, has to be molded to each individual. Every person is so different and requires a unique diet and workout plan.
That being said, let's first discuss what soreness actually is. You might feel it the day after you workout, but typically the worst of it is two days later. When you feel sore, your muscles have built up lactic acid inside. Lactic acid is a byproduct of muscle metabolism. So essentially, soreness caused by working out is called DOMS (delayed onset muscle soreness) and can make your muscles feel extremely tight and uncomfortable for a few days after the workout. Your muscles also become inflamed post-workout, especially if you were doing heavy strength work.
You can still weight or strength train other muscle groups that are not sore while recovering a different one. For example, you can train your arms heavy while only going extremely light on your legs if they are sore.
When I'm training a client, I first ask where they're sore. If they tell me their legs are sore, we will train arms with 5lbs or 8lbs and use those for toning the legs. For almost all of my clients, that weight is extremely light for legs and won't cause any new tears. I like to incorporate the whole body, so while we weight train their arms, we are toning/stretching the legs and also helping to drain lactic acid. By lightly working the legs, we are increasing circulation and bringing more oxygen to the muscles which helps to flush out lactic acid and get the legs healing faster! A little two-in-one action!
You do not want to workout the muscle group that is sore and recovering.
When the muscle is experiencing DOMS, it has microscopic tissue damage (small tears in the muscle) and needs to rest and recover in order to avoid injury and also see new results.
That being said, working out and attempting to create new tears on already damaged tissue will be pretty painful and won't get you the results you attempting to achieve. It is crucial to let the muscle group fully recover before you weight/strength train it again.
See... yes and no!
It always comes down to listening to your body. If the muscles are extremely uncomfortable, stretch them out and increase circulation. Once they are back to normal, you can strength train again. Side note, this blog post is not about protein, but make sure you are consuming your allotted amount. These little tears need to be filled with something- that something is protein!
So just remember, if the muscle group is sore, rest and recover it until the DOMS is gone, then you can strength train it again. In the meantime, stretching and moving it is key!