There are two kinds of zombies.
- The very classical one.
Where the dead rise from the grave. Also largely based on Christian mythology.
Those are very impossible.
- And the "One organism takes over control over another organism".
They are unlikely, but possible.
See zombie-ants. It's a fungus that grows into the brain of ants and controls their behavior while eating then from the inside out.
The concept also featured in "The Last Of Us".
Consensus is that it isn't dangerous for humans, because humans don't give the right conditions for it to grow. That is false security. There are plenty of funguses that adapted to grow on humans.
So, one of those gaining the ability to control our brains, or the brain fungus gaining the ability to grow inside the warmer humans, and hello zombies.
Some say the human brain is too complex to be controlled, but that also is false.
- There are hints that the parasite toxoplasmosis, that controls the brains of mice, also increases risk taking behavior in humans.
- And then there's rabies.
Which is likely the other mythical origin of zombies.
It very much controls your brain. Watch the video documents where they show the behavior of a human suffering from it. All mammals suffering from it are behaviorally so controlled, that it improves the propagation of the virus. Fascinating and frightening. (Get yourself vaccinated)
However, those are very much alive. So, all restrictions of alive things apply.
They might not really act on expectations. Like certain drug addicts they don't topple over when shot, since the pain system also is hijacked. But they bleed to death, a shot in the lung of heart kills them as easily as one to the brain. Even if slower.
There is an additional risk for the fungus zombie. The host of the rabies one has to be alive to propagate. Basically only a bite will do, and dead things don't bite*². But the spores of fungi easily spread through air, and stay infectious for some time. Sometimes decades. So a dead fungus zombie can still infect you, even if the host is long dead.
We have ways around that, tho. Ways that prevents spores to reach a host.
If used correctly. (Stupidity and or negligence might kill you)
Well, and there's the whole that you can't vaccinate against fungi.
*²=But fighting them might result getting infectious material (body liquids) in your eyes, nose, or mouth which also can result in an infection. Or into a small open wound, to small to notice, with the same result. Wear PPE when fighting rabies zombies. And get vaccinated!