In his political pamphlet Reflections on the Revolution in France, Edmund Burke certainly does portray himself as a devoted monarchist. In fact, Burke had such a strong faith in the "natural order of things" that he felt whatever exists, in whatever state, should exist. He believed that human governments would simply evolve into something else if they weren't already the best possible. As a devoted monarchist who believed in natural order, he also felt no qualms about the existence of social class structures. However, he also believed it was the duty of the higher classes to assist those in the lower class, and he believed it was only through that assistance that society could become equalized.
Burke grew up in Ireland during the period of Protestant Ascendancy (1691 - 1801), which refers to a time period in which Ireland was dominated by a few Anglo-Irish landowners of the Anglican Church. Even though Ireland was a separate kingdom, it was controlled by Great Britain, which had separated itself from the Catholic Church. Hence, Anglicans in Ireland held all of the political power, excluding Roman Catholics, who, sadly, made up the majority Ireland's population. In short, only landowning Anglicans were permitted to vote, not Catholics and not Anglican lower class citizens. Since the majority of Ireland's population consisted of the uneducated, impoverished lower class, Burke was well aware of the severe gap in well-being between classes and developed the belief that it was the responsibility of the educated elite classes to educate and improve those in the lower class (Stanford Encyclopedia of Philosophy, "Edmund Burke: Intellectual Orientation"). In his view, it was the role of the landowner to improve conditions of the impoverished by improving his land; it was the role of the priest to educate the impoverished; and, it was the role of the lord lieutenant to benefit the impoverished ("Edmund Burke: Intellectual Orientation"). Burke approved of these sorts of changes within a society; he did not approve of using violence to achieve these sorts of changes because he personally saw just how much further damage violence ensuing from the Irish peasants caused. As Burke phrases it to his friend Charles-Jean-François Depont in what became Burke's pamphlet, "Believe me, sir, those who attempt to level, never equalize."
We can see Burke's views on social classes, especially the need for the higher classes to educate the lower classes, when he protests against the National Assembly having shut down the operation of the French Catholic Church and having seized all of its property. The National Assembly was formed in June 1789 by members representing the commoners of France, called the Third Estate. Catholicism had become seen as contrary to the commoners' interests because, due to the Enlightenment, the commoners valued reason above faith; the Church also owned a large percentage of France's wealth thereby connecting the Church to the noble class, the First Estate.
Yet, in Burke's eyes, shutting down and seizing the Church was the exact opposite of what the commoners should be doing since he saw it was the priests who were able to educate the lower class. Burke phrases his views on shutting down the church in the following:
To observing men it must have appeared from the beginning that the majority of the Third Estate, in conjunction with such a deputation from the clergy as I have described, whilst it pursued the destruction of the nobility, would inevitably become subservient to the worst designs of individuals in that class.
In other words, Burke understands the Third Estate thinks shutting down the Church needed to be done because of its ties to wealth and, therefore, to the noble class. However, in Burke's eyes, the Third Estate made a mistake because, in doing so, they also shut down all methods for the commoners to be educated and helped. Thus, they have unwittingly subjected themselves to being even more subservient than before.
Hence, it's clear that Burke has no objections to class structure because class structure fits natural order. Yet, he does object to oppression and feels it is the duty of the upper classes to rescue the lower class from oppression. It can also be said he further feels that the higher classes are necessary because, without them, the lower classes will have no means of being liberated from oppression; only the educated can help the uneducated improve their lot.