Marxism is a super wide and broad concept to reduce to one aspects of this particular play.
However, if we go with Lenin's tripartite version of Marxism, we see that they are a) critique of political economy, b) philosophy of materialism and, c) socialist politics.
Some of the characters exemplify some of these elements:
Lord Goring, for example, lives above his means, has no intention to work, yet expects to live off his father. His father talks about buying him a seat in the House of Lords, to which Lord Goring scoffs off saying how tedious the House of Lords is "when" they have to work (not that they do anything).
Lady Chiltern is the epitome of charity and social grace. Yet, Lord Goring consistently puts down this "modern necessity" for charity and philanthropy. In fact, this is a topic also present in "The Picture of Dorian Gray", "The Importance of Being Earnest" and "A Woman of no Importance"- the excessive preoccupation of charity for the sake of attaining social standing.
Sir Robert Chiltern and Mrs. Chevely are also an examples of the marxist obsession with materialism and the recklessness of obtaining it by any means, even if the means are illegal. Power is everything.
In all, the major characters indeed represent fragments of the tripartite version of Marxist thought, and it was Wilde's way of criticizing the hypocritical Victorian society.