Cultures cannot suffer. Therefore it is senseless and even harmful to talk about actions that result in harm to or destruction of cultures as immoral. This places a higher value on protecting perceived qualities of an inanimate entity than on preventing the suffering of conscious human beings. There is and can be no innate tragedy in the death of a culture or language apart from the suffering it causes to individual human beings.
The suffering caused by a culture's death or change can happen for various reasons. At the most basic level, adult humans don't appreciate the disruption involved with learning new social norms; it's just a pain. But there are additional and unnecessary causes of suffering that are built into the values of some cultures: namely, an explicit and conscious value that preserving the culture is itself innately good; and that if the culture were to change or disappear, it would be a moral disaster.
What prompted this post was Katja Grace's discussion regarding deliberately bringing up children to speak minority languages, thereby limiting their social contact with the rest of the world, "Agreeable Ways to Disable Your Children". (There are links to far more controversial proposals that have actually occurred in the real world.) By teaching children to be monolingual in an obscure language within a broader nation-state, parents are preserving their language, but limiting (disabling?) their children. Humans do indeed sometimes keep their children isolated in ingroups and one of the best insulation methods is by teaching them only the ingroup's language. This is one example where a moral value about preserving culture is inconsistent with preventing individual suffering. The language doesn't care that you preserved it; if you force yourself to perpetuate, you're just making it harder on yourself, and your children.
There is a more general argument to be made here, that more conservative, less open cultures which most strongly harbor an explicit value of preserving themselves for their own sake are doing something similar to their members by limiting them and making them suffer unnecessarily.
To make this slightly more concrete, imagine two cultures which differ in cultural conservatism. Culture A is a mercantile civilization that is frequently changed by its people's contact with foreign lands; they shrug and adopt the food and ideas of the people they meet. Culture B on the other side of the river is more conservative, with an explicit and conscious moral sense that their culture is intrinsically valuable, and that if it were ever lost, it would be the end of the world. Culture B constantly fights the advance of the new and its people wring their hands and gnash their teeth at the strange food and ideas polluting the next generation. In the end the material conditions of both places are the same, but the people in Culture B have suffered more, and unnecessarily. (If nation-states are cultures, then I would put the U.S. in Culture A's category. We've changed far more than we realize due to contact with other cultures, and I expect in 50-100 years we'll be, for example, far more Asian than we are now. And because we're an A-Culture, it fortunately won't bother us that much.)
On the other hand, culture is not just noise; cultural values can be better or worse in their impact on material well-being, so there is potentially still bad news in culture change beyond just the degree to which the culture causes its members to consciously decry change. But again the change or loss of these values must be evaluated only with regard to the effect on individual humans. It's tempting to object that we're still assuming values here in order to evaluate their worth. But there is a baseline that exists independent of acculturation, and it's composed of animal essentials - food, shelter, sex, and status.
[Added later: The Wall Street Journal publishes this article about an Eyak language preservation enthusiast. To put it bluntly: what is the value of this work, to Eyak-descendants or anyone else? Thanks to Thurston for the pointer.]