What are some lessons, pitfalls, and dangers in reading Chaucer in the 21st century?
I would say that one lesson is that the art of good story telling is just as important as ever. Chaucer's Canterbury Tales is a story about a group of people telling stories to each other. It was a popular read back then. It still is today. But more importantly, people in the 21st century still love getting together and swapping stories. It might not be on the road toward some pilgrimage, but social networking is hugely popular, because it gives people a way to share their "story" or "adventure" or whatever.
I think another lesson is the old adage "don't judge a book by its cover." Canterbury Tales is full of stereotypical character archetypes, but from their stories it's clear that each person is more than the represented caricature. For example, several of the characters that you would expect to tell a wholesome and appropriate story actually tell a story that is quite dark.
The dangers and pitfalls of Chaucer's Canterbury Tales can be that many of his characters tell stories of extreme pessimism and negativity. The wife of bath's story is not a positive outlook on marriage. The miller is an abusive character to start with. Several of the religious characters are corrupt. A reader should be careful to not fall into some of Chaucer's cynicism.