Bunburying has several connotations:
It was said by Aliester Crowley, according to Neil McKenna, that Oscar Wilde once "took the coach to Banbury, met a Public School boy there, and the agreed to meet in Sunbury" for a secret meeting. Hence, the word "Bunbury" was coined by Oscar and his set as a term that means "leading a double life" or "hiding certain secrets".
Bunbury was then the name Oscar chose to name the non-existent character of the invalid that Algy goes to visit each time he also wants to go somewhere else and do whatever it is that he does.
Hence, the connotation is double, as most of the symbols, names, and epigrams said in Earnest are also double entendres for Oscar's own tribulations.
Earnestness, also according to McKenna might be a Victorian code name that is a bad translation from the French of the term "Uraniste" or Uranian (homosexual) Earnest/Uraniste. Taking it from that view, then the entire play takes a different perspective.
Yet, if we use the term "as is", bunburying would be the antithesis of earnestness, that is, lies vs. truth. In those days, however, the truth "was never pure and rarely simple" and the hypocritical, uber prudish and elitist Victorian mentality permeated just about every aspect of morality and righteousness.
Everyone bunburied amidst the societal insistence of earnestness.