If you're trying to get a learning algorithm to do a task that humans can do. And if your learning algorithm is not yet at the performance of a human. Then manually examining mistakes that your algorithm is making, can give you insights into what to do next.
For example: No matter how much better you do on dog images, or on Instagram images. You at most improve performance by maybe 8%, or 12%, in these examples. Whereas you can to better on great cat images, or blurry images, the potential improvement.
What should you do if you find data with incorrect labeled examples?
TRAINING SET
If the errors are reasonably random, then it's probably okay to just leave the errors as they are and not spend too much time fixing them.
They are less robust to systematic errors. So for example, if your labeler consistently labels white dogs as cats, then that is a problem because your classifier will learn to classify all white colored dogs as cats
TEST/DEV SET
During error analysis to add one extra column so that you can also count up the number of examples where the label Y was incorrect