I teach beginning photo classes. I discourage much burning and dodging, but I do show them how to burn edges and corners. Due to the geometry of the enlarger's optical system, the edges receive less light than the center, and the corners even less. To correct this visual issue a standard burn in from the edges is precisely what is needed. It has very little to do with what's in the image; more to do with the evenness of the projected field.

I really don't want them to do much burning and dodging because they often think that they can simply do so, thus avoiding having to learn the basics of print exposure and contrast, and how to use both of these factors together to achieve a print. I have had students who have adopted burning and dodging and end up "painting" the image, exposing every part more or less independently of the other parts. Why not just take a painting course instead? It would be easier and more likely to succeed.

Whenever you have to burn or dodge to "fix" something, you can be sure that there is something wrong with your basic vision and metering/development decisions. The techniques are great for correcting edges, and for making minor adjustments in local values for the purpose of integrating the image. They are not good for covering up your mistakes. Unfortunately, I believe that most photographers, at least beginning ones, use them to try to cover their errors.

I very rarely burn/dodge. I'm not rigid about it, but I just don't think it is generally all that useful except for the uses I've mentioned above.