The main character in my novel is strongly influenced by a character who never actually appears in the novel. (She's part of his past.) Some of my readers object and say I should re-think this and perhaps give them a scene together. Every scene I come up with feels contrived. They just wouldn't cross paths. Are there examples in literature of characters who have a profound influence but don't actually appear in the story?

There’s no need for a character of influence to actually appear on stage in a work of fiction. Just take a look at Vladimir Nabakov’s Lolita. The narrator Humbert Humbert has an overwhelming desire for nymphets, his word for nine to fourteen year old girls he finds alluring. He traces this desire back to an early love cut short when he was about thirteen: Annabel, who was just a few months his junior. The reader gets glimpses of this character in Humbert Humbert’s memory, but that’s the extent of it.

Characters crucial to the action of the story can remain entirely off stage and work effectively. In “A Jury of Her Peers,” a short story by Susan Glaspell, two women accompany their husbands—the police chief and a witness—to the Wright home where Mr. Wright has been murdered. Minnie Wright, the man’s wife, is in jail as a suspect. As the men go through the house piecing together the crime, the women come across evidence of distress and interruption in the kitchen, Minnie’s domain. They find uneven stitches in a hand sewn quilt, a half full bag of sugar next to the wooden bucket of sugar, and the table only half cleaned. The most damning evidence is an empty birdcage and, later, the dead bird with a broken neck wrapped in silk and tucked under quilt pieces in a basket. The story hinges on Minnie’s actions and motivations, yet she never once appears.

So, keeping this character off stage is certainly an option. But you’ll want to make sure you’ve characterized her well enough that the reader understands how and why she has such a strong influence on your main character all this time later.