What works OFFLINE
These systems keep running while you are away:

1) Lumberjacks continue chopping trees
2) Tree planters keep planting new trees
3) Machines in your sawmill continue generating income
4) Samuel will complete recurring orders, if his automation is enabled
5) Workshop and house construction timers continue to progress
6) You receive offline income (with a 50% rate)
7) You also gain village coins offline (within offline limit)

What works ONLY ONLINE
These features require you to be inside the game:

1) Regular sawmill orders are completed only when you are online
2) Forklifts move materials around the sawmill only while you are online
3) Any actions that require tapping or manual interaction
NOTE:
While offline, you receive 50% of your normal income (including village coins).