Does any Christian denomination teach Genesis 3 as a description of human evolution from animals?

(Aside: "evolution" is an overloaded term. On the one hand, natural selection, or the ability of species to adapt within limits to changing conditions is well documented and does not contradict "young earth creationism". In fact, natural selection has been shown to lead to extremely rapid diversification, faster than predicted by many materialist models! On the other hand, the idea that all extant organisms "evolved" from a single, common ancestor is based on "evidence" and assumptions that are highly circumstantial and/or based on philosophical presuppositions. Since your question asks about the latter, I will use the less ambiguous phrase Common Descent in this answer.)

It is quite unlikely that Genesis 1-3 actually corresponds in some manner to the Materialist Common Descent narrative. While the attempt to shoehorn human descent into the Genesis narrative, as described in the Question, may look somewhat plausible, there are a number of issues if one tries to extend this to the greater narrative.

First, according to Common Descent, modern humans have been around for hundreds of thousands of years. Scripture directly contradicts this by giving a year-by-year geneology from Adam to Christ, which can be lined up with known history at a number of points. While the manner of expression allows for some "wiggle room" (usually on the order of a year per generation), there is no plausible way that can be made to reconcile the approximately 4,000 years claimed by Scripture, with perhaps (very generously) a ten percent error margin with the supposed orders of magnitudes difference claimed by the Materialist narrative.

Second, the order of events has significant discrepancies. For example, Materialism places stars before land-and-oceans on Earth, and land mammals before sea mammals, whereas Genesis 1 gives the exact opposite order. (For more details, see here or here.)

It is more likely that the entire Common Descent narrative is simply incorrect. The entire premise is based on an a priori denial of God's existence, and significant challenges have been raised from multiple fields simply against Common Descent. Even allowing for "theistic evolution", there are inconsistencies in the relations between organisms that are better explained by multiple, distinct acts of Creation. Moreover, once one admits to a Designer, the reason for Common Descent goes out the window. If God exists, the philosophical rationale for Common Descent is invalid, and the scientific evidence is inadequate to sustain that hypothesis without the philosophical underpinning. That being the case, it becomes far more likely that humans were specially Created, as Scripture states.

Another aside: what's wrong with Common Descent?

Book can (and have) been written on the general issues with Materialistic Common Descent; notable authors include Douglas Axe, Michael Behe, William Dembski, Michael Denton and Stephen Meyer, and the problems are myriad. However, this isn't the place for an in-depth discussion. (Please use chat for any discussions; there is a standing room specifically for this sort of topic.)

With respect to the specific inconsistencies mentioned above, there are a couple issues:

• Different methods for determining relations give different results. While "lateral gene transfer" has been claimed to "resolve" these issues, LGT has not been demonstrated to permit a high order organism to adapt macroscopic features from another, as would be required to explain a number of the inconsistencies.
• The "Cambrian Explosion" produced many new bauplans in an extremely short (according to Materialist chronology and by "evolutionary" standards) period of time.
• Homology is poorly explained by natural selection, whereas it is an expected feature of design.
• Both theory and actual experimentation strongly suggest limits on an organism's ability to change.
• No known natural law is capable of producing novel genetic information. In general, genetic information decreases over time, as confirmed by the vast majority of observational evidence.

Scripture clearly teaches that God Created each animal "after its kind", and observation, both of the fossil record and of living animals, strongly correlates to animals adapting within the limits of their kinds. If one allows that a Designer (i.e. God) has interacted with Creation in order to bring about life as we know it, there is no philosophical reason to suppose He has done so only once. That being the case, the evidence is more consistent with multiple acts of Creation, which is also what is taught by Scripture... and Scripture teaches that humans were not only a distinct Creation, but a unique Creation which was accomplished in a different manner and with different parameters than the Creation of other animals.

Further reading:

• Troubles with the Tree of Life
• Testing the Orchard Model and the NCSE's Claims of "Nested Patterns" Supporting a "Tree of Life"
• Is the Evolutionary Tree Changing Into a Creationist Orchard
• The Top Ten Scientific Problems with Biological and Chemical Evolution (See particularly Problems 5-8)

While I'm not aware that any Christian denomination specifically endorses the theory of evolution, most of them welcome members who accept the theory in one form or another. Probably the clearest statement tending toward an acceptance of evolution comes from the Catholic Church. According to a study by the Pew Research Center:

In 1950, in the encyclical “Humani Generis,” Pope Pius XII said that Catholic teachings on creation could coexist with evolutionary theory. Pope John Paul II went a bit further in 1996, calling evolution “more than a hypothesis.”

According to a Pew Survey, Evangelicals tend to be the most skeptical Christian mainline group regarding evolution, surpassed only by Latter Day Saints and Jehovah's Witnesses. Denominational percentages favoring some form of evolution are as fallows:

• Catholic 58%
• Orthodox 54%
• Mainline Protestant 51%
• Black Protestant 38%
• Evang. Protestant 24%
• LDS 22%

On the skeptical side, both the Southern Baptist Convention and the Lutheran Church-Missouri Synod specifically reject the theory of evolution.

Summary: while no major denomination specifically endorses evolution, most no longer make anti-evolutionism an article of faith. Most Christians in the United States accept the idea of evolution is some form as being compatible with the idea that God created the universe and humans,

Short answer: No.

For the Liberal denominations, see Epimanes's answer on the ECLA: Those denominations see Genesis 3 as a product of its time, allegorically teaching spiritual truths by concepts understandable to their contemporaries. Since people back then did not believe in evolution, it is impossible to suppose that any text authored by and written for them is describing evolution.

For theological conservatives, it's a little harder to find anyone who believes in evolution (But we do exist! Though perhaps I am not the best suited to answer this question, as I believe that Adam and Eve were specially created and not born in the normal way): The story of the fall in Genesis 3 is critical to our understanding of the Gospel, and it cannot be taken as simply figurative without undermining the whole truth of our faith. In Romans 5:12-19, Paul makes a pretty plain comparison of Adam's failure and Christ's success. I honestly can't fathom how one could understand these verses without taking as assumed a historical fall of Adam. Furthermore, the text of the Bible smoothly transitions via genealogies from Adam through the myriad clearly historical narratives, right down to Jesus himself. It certainly makes no sense to have a genealogy presenting historical people descended from figurative people. I point the reader to Chapters 15 and 24 in Wayne Grudem's Systematic Theology, for more information.

Speaking specifically about the ELCA, they have endorsed (macro) evolution. Here's some of their quotes:

Concerning evolution the March 2009 issue of The Lutheran made the following observations:

Lutheran scholars have been leaders in “biblical criticism,” in understanding the Bible as a dynamic collection of stories and documents that evolved over centuries. It’s a collection that reflects the varied experiences and outlooks of its people and writers. It isn’t all to be taken literally, and the use of terms like “legend” and “myth” may be helpful.

Consider the opening chapters of Genesis. According to the Creation Museum, there is one creation story that was directly revealed from God and must be taken literally. The Lutheran approach acknowledges that there are two creation stories in the first three chapters of Genesis, each from different times and experiences and not in agreement on all points. The stories are recognized as filled with symbolism and meaning that have continued value for Christians. ...

Lutherans (in the ELCA, at least) take a conciliatory view of evolution and encourage conversation, not confrontation. Since the Genesis stories do not present a factual account of creation, they do not contradict scientific views of how the world evolved. The common scientific view that humans evolved from ancestral primates about 130,000 years ago doesn’t conflict with the belief that humans have a special place in creation—or that God was involved or is involved in the continuing process. ....

Going to the Creation Museum is a memorable experience, but not for everyone. ...Be alert to the misinformation presented in the displays—including the one that claims the support of Martin Luther, who Lutherans know was definitely not a biblical literalist. (p 28-29)

Likewise, years ago, this was a post on the ELCA website:

What does the ELCA teach about creation vs. evolution?

Creation vs. Evolution

The ELCA does not have an official position on creation vs. evolution, but we subscribe to the historical-critical method of biblical interpretation, so we believe God created the universe and all that is therein, only not necessarily in six 24-hour days, and that God actually may have used evolution in the process of creation. In fact, to deny the possibility that evolutionary processes were used is seen by some as an attempt to limit God's power.

"Historical criticism" is an understanding that the Bible must be understood in the cultural context of the times in which it was written.

The ELCA clearly leaves the door open for (Macro) Evolution.