As an Educator has already noted, “literature” is an expansive field. The relationship between natural objects and cultural objects is likely to vary depending on the work of literature that you’re inspecting.

With that being said, it’s not impossible to draw some tentative conclusions about the relationship between the two. Let’s look at Kate Chopin’s novel The Awakening. As you might know, The Awakening centers on a married woman named Edna and her growing need for freedom and independence.

You could argue that Chopin highlights Edna’s need for liberty with her determination to learn how to swim. Edna practices swimming by herself in what appears to be a sea. You could claim a sea is a natural object.

You might also argue that Chopin highlights Edna’s blooming independence with a hammock. In chapter 11, Edna lies on a hammock outside. Her husband asks her to come inside. Edna, showcasing her autonomy, turns down his request and stays outside on the hammock. You could argue that the hammock is a cultural object. It’s not a natural object. People made it. However, the cultural object seems to serve the same purpose as the natural object: they both point to Edna’s developing independence.

In other literary texts—The Scarlet Letter, The Hunger Games, The Adventures of Huckleberry Finn, and The Book Thief, for example—you might find that natural objects and cultural objects tend to work together to highlight respective themes.

Their apparent unity might cause you to question how different cultural and natural objects really are. You might wonder if the binary is necessary. After all, nature is not separate from culture. As the term “nature object” implies, it can be commodified and turned into a thing as readily as any “cultural object.”

I cannot speak to "relations" between natural and cultural objects in literature because those are going to be different in every literary text, with no simple way to generalize relationships, if that is what is meant.  The differences between natural and cultural objects is a topic more amenable to discussion.

It is important to first establish what natural and cultural objects are.  A natural object is one occurring in nature, not made by man, for example, a rock or a tree.  A cultural object is one that is made by man, such as a decorative symbol or a box.

To the degree that a natural object can be found all over the world, it tends to be a more universal symbol than a cultural one is. Rocks are a good example of this, acting as a means of retribution in literature (and in real life) in many different cultures, from the stoning death of adulteresses in some Middle-Eastern and North African countries to the stone-throwing sacrifice at the end of Shirley Jackson's "The Lottery," or conversely, to operate as a positive symbol of building or boundary as in Frost's poem, "Mending Wall," or in the Bible, which is Middle-Eastern in origin. Water is a symbol of healing or journey in literature all over the world, in the Bible, in The Adventures of Huckleberry Finn (Twain), and in mythology all over the world.  Hence, natural objects, as they are used in literature to operate as symbols, tend to be used for similar purposes in many cultures.

Cultural objects that are used symbolically in literature tend to be imbued with meaning that is more culturally specific.  One example of this is the symbol used by the Nazis in Germany, the swastika. The symbol is still used in literature to represent the Nazi or neo-Nazi movement, in books such as Maus (Spiegelman). The symbol, known as a svastika in India and other countries with Hindu and Buddhist populations, is a sacred symbol that represents good fortune. When such a symbol is present in literature from these countries, it is meant to evoke these, not the Nazi movement. The box operates differently as a symbol, too, in different cultures. In "Pandora's Box," an early Greek myth, the opening of the box releases all the evil into the world, along with hope, which is the last thing left in the box.  In Western literature, a box has other kinds of meaning, for example, a representation of constraint, as in Donoghue's Room, in which a mother and child are confined to what is essentially a large box, a receptacle of the unknown, such as the "black box" from an airplane crash, or the black box in Jackson's "The Lottery," which appears as a symbol of death. A man-made object is generally going to be culturally specific, and of course, to understand the symbol, one must know something about the culture.

Geography, which provides all cultures with many natural objects, is the leveling ground for universal or nearly universal symbolism.  Those objects made by man arise out of the culture in which man lives, so their meanings as symbols are far more likely to differ.