A moral duty? Almost certainly no. I can't watch videos at the moment so sorry if I miss the thrust of the linked Youtube video.
We definitely don't have a moral duty to bring back, say trilobites or some other animals that's been extinct for a huge length of time and man had not hand in exterminating.
But what about, say, Steller's sea cow, which was unambiguously our fault and only happened less than three hundred years ago? If we are able to bring back a species and place it in the environment it evolved to be in, are we obligated to do so?
Personally, I'd say doing so may well be a worthwhile scientific project, but not a moral duty. For one, how do we pick which to bring back, and how do we rank their importance? Is it just ease? Or ecological significance? Maybe something else? Say the black rat finally loses its fight against the brown rat and goes extinct. Do we need to resurrect it? Why? Does this line of thought preclude us from exterminating the species of mosquitoes that carry malaria?
Don't get me wrong, I'd be the first in line to see a mammoth if one were cloned. I just don't think this issue enters the realm of morality, and if it does, it's raises a host of new, difficult problems.