Is it safe to leave my propane heater on all day?

Unattended?  No.  The portable ones say right on them, don't leave them running with no one there to monitor them.  If it's a built-in unit or a more comprehensive unit with safety features built in, the instructions may be different but generally speaking, if the instructions say not to leave it running without someone around to make sure everything is okay, then that's the rule.

If you aren't home to attend to the heater, why would you bother heating a building that doesn't have anyone in it?  It won't take as long as all that to get it back up to temperature, and it WILL NOT take more fuel to heat back up than it would use running all day.  Disregard anyone who tells you different, they are mistaken.

why would you waste the propane if you're not home?

If you're there with it, yes. And make sure you ventilate the room.

There are many different types of propane heaters so you will need to be more specific.  just remember that anytime you burn fuel you are using oxygen and adding carbon dioxide, carbon monoxide and nitrogen oxide. it burns rather clean but it still has exhaust.

Not knowing what type of heater you have I cannot give you a safe answer.