We should try and protect the dying languages only if there are historical documents, artifacts, massive literature, or just any other historical and beneficial significance to perserving the language. It doesn't seem feasible to keep every language that ever existed alive. Just as with different species, certain languages will die out. Some of them should definitely be kept alive though.
I think it's important to keep languages alive in some form for future generations to fully understand their native cultures. Many Native American dialects have been lost and scholars and tribal members now have difficulties making sense of their history.
Sadly, many languages are "dead." A language that has no speakers is called either dead or extinct. A language only spoken by elders and not by young people is called "moribound." A language that has very few speakers is called "endangered."
Dead languages include Ancient Egyptian, Ancient Nubian, dozens of African languages, and Native American tongues.
Endangered languages exist all over the world, but include Romano-Greek and Judeo-Italian.
Sadly, sometimes the few remaining speakers do not show much interest in keeping their language alive. In a recent case reported by Time Magazine, the last two living speakers of a Mexican language called "Ayapaneco" refuse to speak, despite the fact that they live within a half a mile of each other. Unless historians and others can convince pair to talk and record their experiences, a language and way of life that existed for thousands of years will vanish forever.
You can read more about the story here:
This is simply a matter of personal opinion.
On the one hand, you can argue that we should try to protect these languages for cultural reasons. A language is a major aspect of the culture that speaks that language. When the language dies, a major part of the culture dies. By preserving languages, we preserve the cultures of those who speak them.
On the other hand, it is most likely impossible to truly save endangered languages without a huge effort. Such languages are typically endangered because their speakers have been assimilated into a larger culture and must learn the language of the larger culture in order to prosper. It is very difficult to get people to spend their valuable time and effort learning a language that will do them no tangible good. It makes more sense for them to learn a language that will help them economically and socially.
So, preserving languages and their cultures is a noble idea, but it is often the case that the young people whose parents speak an endangered language do not have any tangible use for the language and will not really care to go to the trouble of learning it.