The Sawmill is the lumber field. Increasing its level increases lumber production speed and the temporary storage cap held in the sawmill. Production is not auto-banked: once production starts the temp cap is reached in about 12 hours, so lumber must be collected before then -- tap the wood icon on the sawmill or the arm icon on the Warehouse. This 12-hour cycle is a key automation trigger for a daily-collect loop. The Sawmill is a prerequisite: the Archer Camp, Workshop and Trap Factory cannot upgrade, and Keep Lv21 cannot be reached, unless the Sawmill is leveled first. Upgrade cost is Food/Stone/Ore -- the Lumber column is blank (the lumber field never costs lumber). Stone enters at Lv7, Ore at Lv12; Keep-gated 1:1 (Keep n per Sawmill n). Construction-speed example +0% through Lv17 up to +610% at Lv50. No duty-officer section. Admin note: the number of sawmills buildable (its Building Capacity) rises with Keep level -- about 7 at Keep 1, 10 at Keep 6, 15 at Keep 33 (unclear if higher). So both per-building output and count scale with Keep. Lumber upkeep is not discussed (only food upkeep, on the Warehouse page), so lumber production is effectively net-positive.
Original research source: Evony Guide Wiki