It's not about instructors in the way you're thinking. The "instructors" in question are youth and old age. It's saying that old people aren't always wiser than young people because they may have forgotten much of what they've learned earlier in life.

It is easier to understand in context, if you read the paragraphs before and after to get the general sense of what he is saying. It's traditional to say that age brings wisdom. It's traditional to say that young people can learn from the experiences of older people. Thoreau contradicts this. He contradicts it completely and strongly. According to Thoreau, old people have almost nothing to teach. The wisest man has learned almost nothing by living. The old have no good advice to give the young. The reason, he says, is that the experience they've had is "partial," they've only experienced a tiny bit. And he says that their lives have been failures--so a lot of what they think they know is wrong. Simplifying his language a little, he says "I have yet to hear the first syllable of valuable or even earnest advice from my seniors. They have told me nothing, and probably cannot tell me anything useful. Here is life, an experience I have hardly tried; but it does not help me that they have tried it." In your sentence, he personifies "age" and "youth." Has age taught an old person much? Has age been a teacher? Has age been an "instructor?" He says no. "Age" is not "qualified" to be a teacher. Why not? He says most people have been miserable failures(!). Therefore, the things they think they have learned from life are wrong.

Hello

Jonathan's answer is great!