Thank you for your question.
Basically, I would say that Islam and a classless society could exist, simultaneously.
Throughout my studies, I have come to the conclusion that Islam proposes an economic system, which is unique among capitalism and communism. It could be called: an asset-based model of society.
This model has two elements of capitalism: (a) the right to private property; and (b) the right to earn, and keep, as much wealth as one would like.
What differentiates this model from capitalism, is that Islam prohibits usury.
Furthermore, debt is strongly discouraged in Islam. So, the question would naturally arise (from people who invest in interest-based ventures): What would I do with any extra money, if I cannot invest it in an interest-bearing security?
The answer, would be one of three things.
Firstly, an individual, in my opinion, should have enough wealth saved up to provide for one year of living expenses. Actually, the Prophet - peace and blessings of Allah be upon him - did something similar. Though, he had non-perishable food items as well (which are, in fact, a substitute for money, and can be traded themselves). Where I live, one year's living expenses would be about $24,000.
Secondly, we see that Islam advocates an "asset-based" model of economics. Debt, on the other hand, is a liability.
An asset could be defined as something which you own - meaning, that if you were to suddenly suffer a health-related injury, and were unable to earn money from your regular job, you would nevertheless have no fear of losing what you own, because you do not owe any payments on it. (Think, for example, about person who goes into debt for a $100,000 home, or a $10,000 car). So, for the person who has extra money to invest, a next step (after having a year's worth of expenses saved up) would be to acquire assets which are more or less essential to one's way of life - land, housing, and a means of transportation.
I could say more about this subject, but I figured that this would be sufficient by way of an introduction.