I think the question is not if or whether Chaucer's The Canterbury Tales is relevant today but how the poem is meaningful to us today. It is always a fair question to ask if a work written seven centuries ago still speaks to us and, in the case of The Canterbury Tales, what it says to us, but a strong case can still be made that the work is not only relevant but helpful.
Chaucer chose to write The Canterbury Tales in the vernacular. More precisely, he composed the poem in a Middle English dialect probably spoken in and around London, so he clearly wanted the poem to be accessible to the largest number of literate people in England. From a literary and cultural standpoint, Chaucer's decision to write in Middle English, even though we now need a glossary to understand the vocabulary, is very important because readers can trace the development of English language and culture through Chaucer's poem, language that bridges early Middle English to the beginning of Modern English represented in Shakespeare, for example. As a historical artifact, the poem is priceless.
Perhaps even more important from a cultural standpoint, Chaucer's use of twenty nine "sondry folk" who have been brought together "by aventure" (the desire to make a pilgrimage), allows a wide spectrum of high and low, good and bad, religious and irreligious, intelligent and really dumb people to tell us what their lives are like, what they think is important (as exemplified in their tales), and what the world in England was like at the end of the 14th century. There are no comparable works in English or other European languages that encompass, in one work, all levels and walks-of-life. The question is, "Does that make the poem relevant for us now?"
For several important reasons, yes. George Santayana, the mid-20th century philosopher, is perhaps most well-known for his comment that "those who cannot remember the past are doomed to repeat it," a belief that has some connection to The Canterbury Tales and many other works of literature, history, and philosophy. By looking into the lives, intellect, and beliefs of a representative number of people who lived over 700 years ago, we are able to see that, over time, humans have not changed dramatically. Language changes, clothes are different, political and religious life is dramatically different, but human behavior and ways of interacting with the world remain relatively constant, a fact that is not only comforting but also tells us human beings have remained constant in fundamental ways. Some of us might conclude that the constancy, rather than institutions, is the most important part of life. In simple terms, we would not know that, perhaps not even suspect that, without having The Canterbury Tales (and other works) to help us measure ourselves against those who came long before us.
This constancy is perhaps most clearly depicted in the opening lines of the General Prologue:
Whan that Aprille with his shoures soote/The droughte of March hath perced to the roote,/And bathed e'ery veyne in swich licour,/Of which vertu engendred is the flour. (ll. 1-4)
When April with his sweet showers/has pierced the drought of March to the root/And bathed every vein [of the plants] in such liquor/that it brings forth the life of every flower. (my translation)
Aside from the fact that Chaucer recreates a beautiful picture of the return of spring, appealing equally to those of the 14th century as well as the 21st century, this scene's beauty lies in its permanence. Like the behavior of the pilgrims, which can be found in the people of each intervening century, Chaucer depicts a scene that has played out every year for the last seven centuries, a reminder of our connection in time and space with those who went before us.
This does not answer the question whatsoever.