Mr. Feeney is named after Cheryl's boyriend David Feeney, who she was dating (and presumably living with) at the time. Cheryl has stated that she and her boyfriend had played with him all day when they first brought him home.
Mr. Feeney died in his litterbox on the same day that Jim, Andy and Dana were going to a Chicago Bears game. Jim at first agreed to bury him before he went to the game but decided to leave him in the freezer they had in their garage so that he could got to the game. When Cheryl found him accidently, she became infuriated with him. That night, Jim payed tribute to Mr. Feeney by playing "Amazing Grace" on his harmonica, at which point, Cheryl forgave him.
Mr. Feeney is named after the executive sotry editor David Feeney, also the name of Cheryl's ex.