Cooking might be enough to cause the agents used for euthanasia (barbiturates, mostly) to break down, though I’m not sure about that. Depending on the amount used and the size of the animal, you’d expect fairly low amounts of barbiturates in any amount someone might eat, and the agents would probably accumulate more in fatty tissue than in muscle. It’s possible, though, that you might consume an active dose of barbiturates, which could pose a problem if you plan to drive afterwards, especially if you consume alcohol.
That being said, the reason why the animal was put down may render it unsuitable for consumption. Cows that have been put down, for example, are unfit for human consumption. And a lot of people, including myself, would argue that any animal that’s put to sleep at a vet’s office is inappropriate for human consumption, at least in our culture. If you live somewhere where you find vet’s offices and pets, you probably live somewhere that has supermarkets, and so you couldn’t argue desperation.
Cows are generally aren’t put down with euthanasia. They’re generally shot, whether with a humane killer or a regular rifle. They’d be safe to eat if killed by those means, but by the time they’re sick enough to be killed rather than processed, they’re either so full of different drugs from treatment or so wasted from being ill that it’d be pointless to butcher them.
The only exception to this would be a cow that was put down because of a broken leg or injury-related paralysis. Those are generally picked up by the “dead truck” though, going towards pet food.
There was the recent story about the elk (moose?) that got its antlers tangled in electrical wires. It was still alive when they got it down, but IIRC it was injured and they decided to euthanize it. They mentioned that the carcass was disposed of rather than butchered and eaten because of the euthanizing drugs.
This doesn’t seem to be the case. The FDA investigated traces of euthanizing chemicals in pet food to determine where they came from (they were worried, it seems, that pet food manufacturers might be using put-to-sleep cats and dogs for kibble.)
It turned out that the chemical was from dead horses and cattle which were euthanized, and then rendered into pet food.
This page states that the amount of the chemical is unlikely to harm your pet.
However, if, God forbid, you should ever have to put your pet down, some have suggested that the trace amounts in your pet’s body could cause them to suffer unecessarily. The FDA’s report does not address this concern.
My boss told me about a previous shelter she worked at, in which someone brought them a couple of perfectly healthy goats and requested that they be euthanized. The shelter did it, but then the person asked for their carcasses back, and the shelter refused: as near as they could tell, this creep was using the animal shelter as a cheap way to slaughter his livestock, so he wouldn’t have to pay a slaughterhouse to do it. And that’s not the role of a humane society.
So there are some people out there who’d want to eat an animal that was full of sodium pentobarbitol.