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 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.

It's not only possible, it has been documented, in humans, within the past century. Google for the "blue Fugates" -- they were an inbred family/community in Kentucky who, due to a mutated gene, had a much higher than normal level of methemoglobin. This altered hemoglobin doesn't carry oxygen efficiently, but it is quite blue in color -- blue enough to overpower the normal pink skin color in Caucasian humans.

The belief is that the Fugate family had a couple members who were born with this mutation, and as a result they were shunned by others. Add this to geographic isolation, and you get inbreeding. Over a period of a couple centuries, this led to everyone in their extended family (which was their entire community) having blue complexion. After the family broke their isolation in the late 20th century, physicians found a simple treatment: injection of methylene blue dye converted the methemoglobin to common hemoglobin, and their skin color changed from blue to pink in minutes. Periodic treatments are needed, and the mutation is still present in their family/community, so the Blue Fugates aren't gone, they're just hiding among us.

Vanabins

image of ascidians: http://frontiersmagazine.org/post-11/

Amavadin: from https://en.wikipedia.org/wiki/Amavadin

Vanabins are vanadium containing molecules found in sea squirts and some other organisms. Depicted is amavadin which is from a mushroom but which I think is more hemoglobin-like than the true "hemovanadins" from ascidians. They were called hemovanadins because they were thought to participate in oxygen transport but according to what I read that is now in doubt because these creatures have hemocyanin too. Oxygen transport still seems likely to me.

In any case: they are hemoglobin analogs, they can do oxygen transport and they are awesome colors of blue and green. Hemovanadin blood could be a fine blue color.

But why would a mammal use vanadium for oxygen transport when iron works so well? What if the iron brought trouble with it? That is actually the case for mammals - infectious organisms also want iron. When you are infected, a molecule called ferritin grabs all the iron it can, denying it to the infection.

Immune response Ferritin concentrations increase drastically in the
presence of an infection or cancer. Endotoxins are an up-regulator of
the gene coding for ferritin, thus causing the concentration of
ferritin to rise.... Thus, the iron stores of the infected body are
denied to the infective agent, impeding its metabolism.[24]

A consequence of this in real life is anemia - bound to ferritin, iron is denied to the red blood cells too.

What if there were some prevalent infection which depended on an organisms iron? An organism with minimal iron would have resistance to that infection. If infected, it would not have to get anemic because its cells are using V not Fe. Even if vanadium is less efficient at oxygen transport, the disease resistance conferred could cause hemovanadin-based blood to spread throughout the population.

How could mammals get hemovanadin? The sea squirts are our distant ancestors. Perhaps the gene for hemovanadin is still in mammalian DNA, sequestered and unused in some dusty corner of the genome. An accidental mutation restores it, and it is used shoulder to shoulder with iron hemoglobin, conferring advantages. Later an organism mutates out the hemoglobin gene leaving only hemovanadin - with consequent improved fitness.

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.

I know you said blood, but it is documented that colloidal silver taken as a supplement causes argyria which turns the skin blue. Colloidal silver is used often to sterilize water among other things if this helps any.

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.