Ever since I found out about the green blooded skink, I have wondered if it's possible for other blood colours to evolve among vertebrates, specifically blue in mammals.

I can't see why not. Octopi have blue blood. Apparently its due to a protein called Hemocyanin that binds with copper. So its clearly physically possible in complex Earth life.

For endotherms, iron is in the heme groups binding oxygen, yielding a reddish color.

For ectotherms, copper is in the heme group binding oxygen, yielding a bluish color.

In order to bind oxygen, each protein chain binds to one heme group, allowing a maximum of four oxygen molecules to bind per one hemoglobin molecule.

At heme's center sits an iron molecule. The iron makes heme look red-brown. But what if the iron is swapped for a different metal?

...in cold-blooded animals, blood appears blue because copper atoms sit at the center of the ring and bind to oxygen.

link

It is probably a good bet that binding affinities for the iron-heme-O2/CO2 complex at 98F are better attuned to optimal gas exchange relative to the copper-heme-O2/CO2 complex, since this is precisely the substitution which occurred during mammalian evolution, thru many intermediate steps, doubtless, including modification of the carrying protein/heme platform.

Which means blue blood is not favored relative to red blood, for mammals.

Impossible though? Such an experiment could perhaps be run by depleting iron and raising the availability of suitable copper sources. That would be something for a molecular biologist to grapple with. I expect toxicity would likely be a large experimental hurdle, the process likely including reactivation of hemocyanin biosynthesis, per Molot.

I am going with probably not

This is entirely possible in the sense of why not. Its not like blood color is a factor in its function, merely a byproduct. Its just our evolution resulted in iron based hemoglobin so walla red blood.

Here's why I say probably not:

We and most mammals are pretty complicated organisms. Blood is a pretty basic functioning trait. Its kind of unlikely that evolution would seek to revise our blood color without a strong reason. Some would hat wave mating for this but I feel its pretty safe to say blood isn't going to play out in most mammals mating habits unless vampires become a thing.

Maybe our nature can find improved immunity function that changes its color. And this is really the complicated part. Finding some adaptation that requires significant change in the blood chemistry to result in change of the color.

It is possible. Hemocyanin would be what you would use and not hemoglobin. As stated, hemocyanin is worse at oxygen transport, compared to hemoglobin. But not all hemocyanins are the same, there are more effecient versions with better cooperative binding.

If you go with methods of directed evolution, we use to produce better enzymes, you could hope to produce hemocyanin, that would be good replacement for hemoglobin.

That for sure would take a lot of changes in your body to make it work well after that. But it is possible.

I want to support @anon's answer of

Probably not

and substantiate their assumption with scientific fact:

Hemoglobin (red) is better at transporting oxygen at the body temperature of mammals (around 38°C). It would be an evolutionary disadvantage for a mammal to have hemocyanin, because it would suffer from permanent "lack of oxygen" relative to other mammals with hemoglobin. Think of the blue-blooded antelope being out of breath more quickly than the red-blooded antelope. The lions would eat all the blue-blooded antelopes first.

Hemocyanin is better at transporting oxygen at the low body temperatures that ectotherms often have to function with. Blue blood is an evolutionary advantage for animals with low or changing body temperatures.

It has been suggested that warm-bloodedness helped mammals (and birds) better cope with fluctuating temperatures during Earth's history. If that is correct, cold-bloodedness (and blue blood) would have led to the extinction of mammals.

Other sources:

• German wikipedia articles on body temperature and hemocyanin.