Fuel is necessary to process materials in a Forge. Fuel has to be applied to each single forging process (smelting, hardening, etc.) in order for the Forge to work. You cannot place forgeable materials into the queue of Forges without also inserting the appropriate Fuel for each forging process in advance.
Fuel comes in 4 tiers/heat levels that burn with different speed.
Examples for Tier 1 Fuel:
2 blocks of uncorrupted Wood, 2 Logs of any kind, 2 blocks of Snowy Elderwood Leaves, 4 blocks of all other uncorrupted tree Leaves, 8x wooden building blocks like Wood Walls or Wood Floors, 8 Wood Stairs, 8 Haunted Stairs, 8 Carpeted Haunted Stairs, 8 Wood Ladders, 8 Wood Windows, 8 Barred Wood Windows, 2 wooden Beds of many kinds, 2 Wood Doors of many kinds, 2 Wood Tables, 1 Wood Sword or 2 White Wood Tables, 4x Shrubs, 4x Tallgrass, 4x Savannah Tallgrass, 2x Shorewood Husks, 2x Vines, 4x Weeds, 2x Moss, 8x Wood Rods, 4x Wood Slabs
Examples for Tier 2 Fuel:
Examples for Tier 3 Fuel:
Examples for Tier 4 Fuel:
You can read more details about Fuel in the dedicated article Fuel.