The wake is described from the point of view of the character called Peter Ivanovich, who was a friend and colleague of Ivan Ilyich. The only comical elements are concerned with Peter Ivanovich's awkward behavior at the wake. He really doesn't know how one is supposed to behave, since he hasn't had much experience with such things. He knows he is supposed to make the sign of the cross on his forehead, shoulders, and chest, so he keeps doing that until he begins to feel ridiculous. Then Ivan Ilyich's widow asks him to talk with her in private in the drawing room. She pretends to be grieving, but it turns out that she only wants to question him about the possibility of her getting any more money from the government in the form of a larger pension or by any other means. Her husband was a highly placed judge, and she obviously believes she should have more compensation for all his years of government service.
The most comical element is in the description of the already awkward, embarrassed and uncomfortable Peter Ivanovich's taking a seat on a sort of cushioned seat described as a "pouffe."
When they reached the drawing-room...they sat down at the table -- she on a sofa and Peter Ivanovich on a low pouffe, the springs of which yielded spasmodically under his weight....on her way to the sofa the lace of the widow's black shawl caught on the edge of the table. Peter Ivanovich rose to detach it, and the springs of the pouffe, relieved of his weight, rose also and gave him a push. The widow began detaching her shawl herself, and Peter Ivanovich again sat down, suppressing the rebellious springs of the pouffe under him. But the widow had not quite freed herself and Peter Ivanovich got up again, and again the pouffe rebelled and even creaked.
This little scene seems worthy of the great Charlie Chaplin. What makes it especially funny is that Peter Ivanovich has to keep pretending to be very serious and sympathetic, while Ilyich's wife has to keep pretending to be grievinig and not noticing the problems her guest is having with the pouffe.
Tolstoy probably wanted to start off this long narrative with a comical and satirical tone because it would work as a contrast with the growing grimness and eventual horror of Ivan Ilyich's increasing illness, his gradual loss of hope, the slowly intensifying pain, and the man's final death. Throughout the story there is a marked contrast between the behavior of the living and the feelings and perceptions of the one man who is facing death. The living persons, including the various doctors and Ivan Ilyich's own wife, seem phony and ridiculous. The only truthful person, besides the dying man himself, is the servant named Gerasim, who really commiserates with his master and doesn't try to cheer him up or to persuade him that there is any hope for him being cured of his fatal disease, which is never named explicitly but appears to be a form of cancer.
Another reason that Tolstoy might have decided to inject some comedy into the early scene at the wake could have been to lure the reader into the story while saving the gruesome details of sickness, suffering and death until the reader was fully absorbed in the plot and fully identified with the dying man.
"The Death of Ivan Ilyich" is a masterpiece of storytelling, arguably the greatest work in Russian literature.