Will God at some point not use a Christian (put them on a shelf) that is entrenched in apathy and sin and continues to do so year after year? Yet believes they are serving God? Living one life at church and another at home.
So this is a tough one to answer. Will God ever NOT use a Christian? If we stop there, my answer is “no.” God’s never done with His children.
Now continuing with the remainder of your question, I don’t know if a true follower of Christ can be “entrenched in apathy and sin and continue to do so year after year.” Especially if they think “they are serving God,” but are living in complete hypocrisy (“living one life at church and another at home”).
True followers of Jesus are not comfortable with their sin. They don’t justify it. Their sin bothers them. It grieves their heart. So to go on in the same apathy and sin year after year without any struggle, without any attempt--by God’s grace--to defeat the sin, and to do so all while thinking that you’re serving God...sounds more like the religious Pharisees that Jesus rebuked in the Scripture. These men were completely unaware of their sin and hypocrisy all while thinking that they were serving God as they had Christ murdered.
So does God give up on using a Christian for His glory? Never. Are some people deceived into believing that God looks at their apathy and sin and thinks, “No big deal?” Absolutely.