Ultimately all life will disappear due to the heat death of the universe so why bother.
Humanity is fairly new, I doubt we'll make it to the heat death of the sun. Modern humans are 300 000 years old (anatomically, 50 000 behaviourally), the sun will run for another 5 000 000 000. We're running out of easy to use resources after not even 150 years of industrial progress, people are getting less and less healthy, fertility is going down thanks to pollution, many eco systems are on the verge of collapse with unpredictable consequences for us, &c.
I think we have many things to focus on before saving animals from the heat death of the sun
> We're running out of easy to use resources after not even 150 years of industrial progress,
All of human civilization (and some pre-civilization time) has been industrial progress, and even if you are counting from the beginning of the popularly-named Industrial Revolution, that's fairly arbitrary but also 260+ years, not 150.
You are assuming it is actually possible for human life to move to other solar systems.
It's much more likely that we could send bacteria or extremophiles to another planet and have them become self sustaining there than it is for a colony of humans, given everything we know today about the limits of space travel (time scale VS resource requirements).