Well, first let us look at what "mono no aware" means as a concept, and then we can see how it fits into The Tale of Genji. "Mono no aware" is a Japanese term that has the literal meaning of "the pathos of things," but it can also mean having empathy because of the transience of the world. It is more of an Eastern concept than a Western one and, as such, can be found easily within The Tale of Genji.
In The Tale of Genji, it is this subtle reality of life and the sadness of passing time (mono no aware) that is the central feeling the reader should come across. Genji (the main character) shows the aristocratic life of the Japan of the twelfth century. We can see some of the mono no aware characteristics even early in Genji's life when he loses his mother at three years old. More subtle reality sets in as Genji receives a demotion and takes a concubine. Much of the sadness here can be seen in Genji's quest for love, taking many wives in search of someone anything like his first concubine.
Probably the most famous chapter that exemplifies mono no aware is the chapter called "Illusion." In this chapter, Genji becomes tired of looking for love through his political promotions and demotions and begins to contemplate how transient life really is. Immediately following is another chapter called "Vanished into the Clouds," which is blank and considered to be the evidence of Genji's death and the reason why the reader should continue to contemplate the truth behind the concept of mono no aware.
In conclusion, it is important to note that the "mono no aware" concept found within The Tale of Genji is also relevant considering that this novel is known as the first "psychological novel," especially in regards to its characterization. Genji is the focus here, of course. The subtle sadness of his life in search of true love and his eventual death should cause the reader to meditate on the truth behind the Japanese concept explained here.