Is there a word for someone who does no work of their own, but criticizes the work of others?

There is no single word that means lazy and critical at the same time. There are quite a few non-specific words though, such as idle and indolent.

Perhaps one of the several types of personality disorders might fit, and a psychiatrist would certainly find the right word for her.

Though it isn’t a single word, armchair quarterback comes to mind. Related terms are armchair caddy and back-seat driver.

One who avoids their own duties is often called a shirker. Oxford Dictionaries Online defines shirk as

Avoid or neglect (a duty or responsibility): their sole motive is to shirk responsibility and rip off the company

The deprecation of those who actually do the work could be characterized as disparaging

Expressing the opinion that something is of little worth; derogatory: disparaging remarks about public housing

Often such disparaging remarks contain an air of superiority. This could be described as arrogant

Having or revealing an exaggerated sense of one’s own importance or abilities:

he’s arrogant and opinionated

a typically arrogant assumption

Your colleague could be described as an arrogant shirker.

I like four-flusher to describe this woman because she is clearly one card short of a betting hand, and her criticism is bluffing competence: ^{(text copied from www.vocabulary.com)}

Anyone who plays poker knows five cards in a single suit is called a flush. What happens if you have only four of that suit? A whole lot of nothing, which gives us the word four-flusher, a person bluffing at cards, or someone who seems promising, but isn't.

Use four-flusher as a synonym for conman, trickster, or fake. You won't hear it used very often these days, but hop in a time machine (a four-flusher will sell you a ticket), travel back to the Old West, kick open the bat-wing doors of the local saloon, and you might hear four-flusher slung across a card table as a brawl gets underway. You might even hear it used as a verb, as in, "You better stop four-flushing, or I'll run you out of this town on a pole."

The word flush seems to be particularly appropriate, although it is a homonym.

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Hypocrite, as suggested by Nicole, is a more conservative descriptor in the sense that this person pretends to be an expert with her scathing criticism, but never demonstrates her expertise with any real productivity: ^{(text copied from dictionary.reference.com)}

noun

1. a person who pretends to have virtues, moral or religious beliefs, principles, etc., that he or she does not actually possess, especially a person whose actions belie stated beliefs.

2. a person who feigns some desirable or publicly approved attitude, especially one whose private life, opinions, or statements belie his or her public statements.

It is particularly appealing to me because the etymology traces it back through play-acting and giving an answer, to sifting: ^{(text copied from www.etymonline.com)}

c.1200, ipocrisie, from Old French ypocrisie,

from Late Latin hypocrisis,

from Greek hypokrisis "acting on the stage, pretense,"

from hypokrinesthai "play a part, pretend," also "answer," from hypo- "under" (see sub-) + middle voice of krinein "to sift, decide" (see crisis).

The sense evolution in Attic Greek is from "separate gradually" to "answer" to "answer a fellow actor on stage" to "play a part." The h- was restored in English 16c.

She's good at sifting other peoples work, but she couldn't produce something sift-able if her life depended on it. If she really were an actress, she'd have to quit and become a critic.

Quack per Oxford Dictionaries Online:

noun

1 A person who dishonestly claims to have special knowledge and skill in some field, typically medicine:

Probably from the primary definition of quack, again from Oxford Dictionaries Online:

noun

The characteristic harsh sound made by a duck:

Lots of irritating noise, but no real benefit. Be sure to do your best at imitating the the actual sound per this YouTube clip:

Well this ungrateful lazy is a nagging-do-nothing (nagging-do-nothing);

Nag; to annoy by persistent faultfinding, complaints, or demands.(http://dictionary.reference.com/)

DO-NOTHING
(Noun) someone who is lazy; (adjective) not willing to work, help, or failing to achieve or do anything important. Synonyms: couch potato, deadbeat, drone, idler, layabout, loafer, slouch, slug, slugabed, sluggard. (merriam-webster.com)

Harridan comes to mind. From Cambridge Dictionaries Online:

an unpleasant woman, especially an older one, who is often angry and often tells other people what to do.

It doesn't fulfill the laziness aspect, but she must do something, even if its work hard at looking like she's working!

I was actually looking for the same thing when I came across this question. I also found something similar on Reddit:

WTW for someone who does not contribute but continues to enjoy the benefits

In there, someone gave the following suggestions:

Freeloader? Leech? Moocher? Slacker?

The English language is just waiting for a good word to describe this concept.