I'm not seeing any of the other answers address anything outside of the western anti-communist understanding of what communists advocate so I'm going to give this an answer even though others have.
I'm going to start off by outlining what I am referring to when I say "communism" because there is not a concrete understanding of what communism is in most countries. Just in case someone who is under the impression that Obama or Bernie Sanders are dangerous Marxists stumbles across this answer, I'd like to make some things clear.
What is communism?
Firstly, I must point out that communism is not something which can be put into place immediately and tested. Communism is not a set of policy ideas or a dogma.
Communists want to usher in a stateless society where the means of production are commonly owned and production is for use by the community rather than sale for the capitalist boss.
While I do not want to assert that all communism is Marxist, I will be talking about Marxism because I deem it to be the most developed communist theory.
"Marxism is the method of dialectical materialism, and the application of this method to social life. Marxism is not simply a collection of pre-existing conclusions or "things Karl Marx said", but a way of understanding and effecting social change based on Karl Marx's method."-communism101
Marxists view society as divided into two key classes.
The proletariat (working class) and the bourgeois class (capitalist class.)
It is important to understand that class is not qualified by the amount of money controlled by an individual, rather how they relate to productive forces. If one sells their labour (when they work in a factory for instance) than they are a proletarian, if one sells products or services for their living they are a member of the bourgeois class.
Communists want to end the exploitative relationship between the workers and the people who own the means of productions (the relationship between the proletariat and the bourgeoisie). When a worker creates a product they are not paid the market value for the product they create, they are paid a wage and the capitalist who owns the enterprise they work in makes the surplus value. Meaning; the capitalist makes money off of what the worker produces.
This relationship also leads to other bad economic conditions like production for sale rather than production for use.
These are just the basics, they are well explained in 25 short questions in "The Principles of Communism" by Freidrich Engels.
Can communism work in a post scarcity economy?
This isn't anything you are going to get a definite answer for.
Many people think that communism will never work due to the abstract concept of "human nature" and many people think communism can work without a post scarcity economy.
Some Marxists explain we are already living post-scarcity, although I imagine the usage of the words differ from yours. Marxists see the elimination of artificial scarcity which plagues modern markets as an important benefit of the transition to socialism and eventually communism.
Can communism work in a post scarcity economy? We can't know for sure. However I can point you to some resources which might interest you.
Many people categorise the economy in the Star Trek TV show to be tinkering with the concept of post scarcity socialism, because human labour is substituted with energy and that ushers in a production for use and the abolition of the worker boss relationship.
Sorry if my answer is not as conclusive as you like, but it is not very answerable as none of us will know what the future holds.
A similar question was also asked on /r/futurology if anyone is interested in that.