I think the typical zombie walk tries to convey a sense that the zombies are no longer human and that all of their physical forms are no longer fully functional.
It seems that the limp goes back even to the very earliest movies
and I don't think I can recall too many movies (I Am Legend being an obvious sample of the opposite) where they didn't limp.
It's true that zombies do not move fast or agile.
In most zombie culture they seem to be far more deadly in the fact that they never sleep, never stop moving in their pursuit of the living, while the survivors have to sleep, have to stop to eat and rest, which is when the zombies usually catch up and assimilate their victims.
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The slow, shuffling gait associated with classic zombies both adds an element of suspense (zombies pursuing a cornered victim give the film-makers plenty of time to squeeze out every drop of drama from the scene, as the monsters slowly close in), and allow a clear identifier to distinguish zombies from the uninfected without a heavy reliance upon expensive special make up effects.
From an in-story perspective, it would be expected that most injuries would be to the legs and feet.
Zombies are defined by single-minded pursuit of prey, with complete disregard for pain or injuries, as well as overall poor motor function.
Fully healthy humans are capable of stumbling, tripping, and falling even under good circumstances.
Zombies would be far more prone to such falls, resulting in a much higher incidence of ankle sprains, broken bones, and lacerations to the feet.
It also seems unlikely that a zombie would be able to tie his shoes, so any person who is not wearing footwear at the time of zombification, or which subsequently lost footwear, would shuffle around in bare feet, and would also not make any effort to avoid stepping in dangerous areas (broken glass, etc.).
Since zombies do not posses the normal healthy healing process, they would form no callouses, and any wound on their feet would not close, resulting in more and more flesh being stripped from their feet as they shuffle along.
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The classical 'Romero-style' zombie moves slowly, almost reluctantly.
It was deliberately created to give them the effect Romero wanted.
Zombies play to the core human fear: the human-looking inhuman thing.
They are so like us in appearance, and so different in behavior.
Their sad appearance - large, staring eyes, obvious wounds, slow motion, inspires pity and/or revulsion.
The way they move was initially intended to seem almost reluctant, as if the people they once had been were trying to resist their body's actions.
The slow motion also mimics many other monsters - notably Michael Myers and Jason Vorheese.
Like the zombie horde, these iconic monsters are slow and implacable.
They kill without remorse, look human but aren't, and you can seemingly escape them - only to turn the corner AND
THERE THEY ARE.
The zombie moves slowly, glacially, maddeningly, implacably.
They stalk humans as Death's harbinger, and they're all the scarier for not needing to move quickly to destroy you.
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I've always imagined it being due to their muscles and such having decomposed some.
I can't come up with any examples but the tv series
The Walking Dead, if you take a look, "fresh" (maybe new is a better world, lol) zombies without any major damages but
a bite or such don't limp, they're just quite clumsy.
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Other than the scary effect, I think their slowness can give the other characters a chance to escape.
If they are fast, what's their difference with other humanoid monster?
Usually zombies are in big troops and their number is plenty.
They are single-minded and don't care for wounds, they could not be underestimate.
Even for characters with guns, they require unlimited supply of bullet in order to defeat the whole troops.
Because the zombies are slow, the best way to due with them is to run.
However, as the scene may not be in an open area, their escape route could be blocked or unavailable.
It is troublesome to due with a hell lots of zombies in a medium-size room like locker room/changing room.
Imagine that they were FAST as normal people.
Think about Half life 2 episode 1.
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I think that zombies are always using full human strength which in turn, deteriorates the tendons and since they're always running after you (in modern instances) that their tendons would be reduced to practically nothing.
This in turn with what everyone else says that a zombie`s muscles are basically rotting away.
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I've noticed this trend that most, if not all, zombies in films tend to limp or walk extremely slowly.  In fact, doesn't this take away from their "scariness" as they aren't fast and don't seem to be agile?  Where did this stem from?
Why do Zombies in films generally have limps?
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The typical zombie walk conveys a sense that the zombies are no longer human and no longer fully functional. The slow, shuffling gait also adds an element of suspense and makes a clear distinction between them and the uninfected. Their slow speed may also allow their prey to escape. The classical Romero zombie was intended to move the way it did in order to create a feeling of conflict between the human mind and zombified body resisting its urges. From a logical stand point zombies were designed to move that way, because it made sense from the persepctive of the logic i.e a decomposing body shouldn't be able to move with agility and it made sense from the perspective of the film i.e creates suspense, gives other characters a chance to escape.
The main use of the slow shuffle is so the audience and other characters can identity who the zombies are without the use of expensive makeup effects. In addition it also gave other characters a chance to escape and even created and extra bit of suspense. Even inuries do not stop zombies from pursuing their prey so from a logical perspective you could assume that zombies move the way they do due to a heavy build up of injury in places like the feet, which causes them to limp and shuffle. The classical Romero-style zombie was designed, in Romero's image, to move slowly almost reluctantly like their human brains were trying to stop their zombified body from pursuing its prey.