Furniture is a new type of upgrade that increases the comfort level of the Oak Inn and increases AP and Gold income for teachers. It also unlocks special effects when teachers collect 3 or 9 pieces of hero furniture. These effect are similar to signature items and can help heroes achieve their top tiers. Furniture comes in different categories and some have a hard asymptotic cap on their yield. This means that spamming the same category can lead to diminishing returns and can reduce the overall efficiency of a hero.
The following table shows the priority for hero furniture sets based on the basic stat rewards and the 2 effects that are unlocked when the hero collects 3 or 9 mythic furniture pieces. Note that this doesn’t mean that a hero is bad if they are lower on the list, as you will be picking up their furniture anyways.