Infurno is a plane of existence located past Purrgatory. It plays a significant part in all three games.
After Love at Furst Bite's best ending scene, Infurno is shown full of the thousand dogs that had just been killed. The ghost cats Coco employed to kill them are also implied to have been summoned from there.
In Purrgatory Furever, Ginger attempts to banish Patches to Infurno, but her attempts are sabatoged by him grabbing Olive and dragging them through the portal with him. The two instead wind up in Purrgatory.
In Patches' Infurno, aside from its use in the title, Infurno is shown to be the ruled over by the diety Bapawmet. They are summoned by Patches in an attempt to get revenge on everyone else by having them kill them. Patches, in a bad ending, can also be seen being chained up in Infurno.