TL;DR; If you cooked the zombie first, you'd probably be okay, but it would be really gross.
The evidence from the comics and shows:
We've seen people eating meat from a person who had been bitten, but hadn't turned into a zombie yet:
Bob: I've been bitten, you stupid pricks! I'm tainted meat.
(Bob laughs as the Hunters spit out their food and begin to retch in disgust)
Martin: Let's just kill him now.
Gareth: No, no, we need him.
Greg: We might as well be eating one of them.
Albert: What the hell's gonna happen? Are we gonna turn? Are we just gonna die?
Gareth: Albert, calm the hell down. We cooked him. Everything is gonna be fine.
Theresa: Why the hell didn't you check him first?
Martin: 'Cause he was fine.
Bob: Tainted meat!
Gareth: Shut up.
Bob: You eating tainted meat!
Gareth: I said shut up!
- The Walking Dead, season 5, episode 3: Four Walls and a Roof
The same scene happened in the comics, although the "tainted meat" was Dale, and the Hunters had a different backstory, and some of them had different names:
In both cases, the victim had been bitten, but hasn't turned; in both cases, some of the cannibals become frantic when they realize their victim was infected, but they calm down when the leader reminds them that:
They don't even know if eating zombies let alone eating an infected human, would be dangerous, and
The meat was cooked.
Unfortunately for us, in both cases, the cannibals are killed by Rick's group before we can see if they will get sick from eating the "tainted meat".
The Well Zombie:
In the episode Cherokee Rose (season 2, episode 4), Maggie tells Rick's group that they can get their drinking water from well #2:
Glenn: Miss, what's the water situation here?
Maggie: Got five wells on our land. House draws directly from number one. Number-two well is right over there. We use it for the cattle but it's just as pure. Take what you need. There's a cart and containers in the generator shed behind the house.
- The Walking Dead, Season 2, Episode 4: Cherokee Rose
Dale and T-Dog are about to get a drink from this well when Dale makes an unpleasant discovery:
They try to haul the zombie out of the well, but it tears in half and its innards gush back into the well; Maggie says her father will seal the well and never use it again. Herschel's reluctance to keep using the water after this incident is probably motivated by two concerns:
It might be dangerous to use the contaminated water
Even if it isn't dangerous, it's really gross.
However, we learn something relevant from this sequence of events:
Maggie: With 50 head of cattle on the property, we might as well be ringing a damn dinner bell.
- The Walking Dead, Season 2, Episode 12: Better Angels
So 50 cattle had been drinking water that had a zombie steeping in it like a teabag from hell for god knows how long, yet they didn't get sick. This might not mean anything - it has been established in-universe (in the comics) and out-of-universe (by Kirkman) that in the TWD universe, animals can't become zombies.
Q: Is there any way we'll see some zombie dogs or other animals?"
A: No zombie animals. Don't these characters have enough to deal with?
- Robert Kirkman, The Walking Dead, Issue #38, Letter Hacks
That doesn't necessarily mean they can't get sick from a bite, but it does mean that the zombie pathogen affects animals differently from humans. Drinking the milk from the cows didn't get anyone sick, either.
Word of God:
Robert Kirkman created the franchise, he writes the comic books and print novels, and he is Executive Producer on both shows. In the "Letter Hacks" section of issue #134 of the comics, someone asked if you could be infected by having sex with someone who had been bitten. Here's what Kirkman said:
If you had a septic wound that was infecting your blood stream, would you pass that infection through sex? No. So... having sex with someone after they’ve been bitten... much like eating a living human’s flesh after they’ve been bitten (show reference... to a thing that happened years ago in the comic) has no effect on you.
- The Walking Dead, Issue #134, Letter Hacks
Kirkman was also asked if consuming raw zombie blood was dangerous:
Q: This is a real Comic Book Guy question – but do I gather that if someone swallows a tiny bit of zombie blood they won’t turn into one of the undead? There was a lot of it being sprayed around this episode [Season Three, Episode 6, Hounded].
A: Yeah, people to a certain extent think of zombie blood as being like the blood from Alien. You know, in the Alien movies it’s like, “Oh god, if it touches you, you explode!” or whatever. Whatever it is that turns these people into zombies is in them already. So the idea of getting zombie blood on your face, which happens all the time, and it turning you into a zombie is something that’s just not the case.
Now, that doesn’t make the zombie bite any less lethal. You know, breaking the skin, having that kind of contact with the toxicity that zombie mouths would have, would be something that causes an infection that definitely would lead to your death and then the thing that’s already in you would turn you into a zombie. So there is a science to this, to a certain extent.
Q: Although, to be clear, you are not technically a scientist.
A: No. No, no, no. But I know about everything that scientists know, I’m pretty certain! But anyway, zombie blood is not quite as deadly as a lot of people think. I wouldn’t drink it in high volumes, though.
- Robert Kirkman, interview with Entertainment Weekly
During season 2, he said the show would explore with this in more depth in the future:
What exactly zombie blood and gore does will be dealt with later.
- Robert Kirkman
Kirkman has also gone on record to explain that the pathogen that makes dead people turn into zombies is not lethal:
The rule is: WHATEVER it is that causes the zombies, is something everyone already has. If you stub your toe, get an infection and die, you turn into a zombie, UNLESS your brain is damaged. If someone shoots you in the head and you die, you're dead. A zombie bite kills you because of infection, or blood loss, not because of the zombie "virus."
- Robert Kirkman, The Walking Dead, Issue #41, Letter Hacks
Just to get this on record once and for all - and it is complicated, I know - here's how zombification works. Whatever makes people come back as zombies after they die - it's inside them. It's inside everyone. No matter how anybody dies, as long as the brain is intact, they turn into a zombie.
So what the fuck does a bite do?
Well, bites, and direct to blood contact with zombie gunk,... causes death. It's a strong infection that leads to fever that kills someone. Then the "virus" or whatever is already in them, turns them into a zombie.
- Robert Kirkman, The Walking Dead, Issue #146, Letter Hacks
To clarify, when Kirkman says "A zombie bite kills you because of infection, or blood loss, not because of the zombie 'virus'", the "infections" he is talking about are mundane, real-world infections1 rather than the specific fictional infection that caused the zombie outbreak.
Greg Nicotero is the special effects mastermind on both shows and has directed several episodes. After the episode Four Walls and a Roof (S5E3) in which Bob, who has secretly been bitten, has his leg lopped off and devoured in front of him by cannibals, Nicotero had this to say about whether you'd be harmed by eating infected flesh:
I don't think so. Since everybody is already infected, I don't think eating ‘tainted meat’ would make that much of a difference... It's great because you really don't know. Since we already know that everybody has already got it, it probably wouldn't have made much of a difference.
- Wet Paint interview with Greg Nicotero
The one instance of zombies being eaten:
In Issue #117 of the comics, we do see zombie flesh being eaten without incident, albeit by a non-human character2:
1: Pathogens that can be spread to humans by bites from animals that eat dead tissue include Pasteurella multocida, Staphylococcus aureus, Staphylococcus aureus, Streptococcus agalactiae, all of which can lead to septicemia. Other pathogens that can cause sepsis include E. coli, P. aeruginosa, E. corrodens, Haemophilus influenzae, other Streptococcus species, Enterococcus species, and Neisseria. Infections like these seem to be what Kirkman had in mind.
2 Ezekiel's claim that the zombie pathogen "seems to have no effect on animals" reflects his own understanding as a character, and isn't necessarily true.