What is the literary function of Mr. Bennet in the story?
Mr. Bennet is a foil of Mrs. Bennet in that he represents everything that she is not: Measured, careless, quiet, and contemplative. Austen uses his characteristics to dramatically set a contrast against Mrs. Bennet, who is arguably one of Austen's most annoying characters with her loud, consistent chattering, her anxiety for marrying her daughters, her exaggerated mannerisms, and her overall insufferable personality.
Within the family itself, Mr. Bennet is simply the head of the family. He shows absolutely no care for the marriage (or lack thereof) of his daughters, and we see how he needed to fetch Lydia from her disgraceful elopement basically to save face. He only showed love for Elizabeth, out of all his family. He was also quite free in calling them names such as "silly."
In all, you could say that Mr. Bennet was the epitome of the happy go lucky man who marries a woman out of social expectation, then realizes what he got himself into, and prefers to withdraw from the family life in favor of a good book and quiet time. When called, he will show up. When needed, he will offer help. Yet, voluntarily? He would not bother much with the Bennet drama.