Can I just run away and pretend nothing happened if things start smoldering around me?

In the Reign of Giants DLC, summer introduces Overheating for the character and for nearby objects as well. If you stay in one place long enough in the middle of the day, things will start smoking. If they're not extinguished fast, they will begin to burn and nobody wants that in Don't Starve.

But is it okay to simply pack your stuff and leave the smoldering objects off-screen fast? Will they not keep smoking and eventually burn?

One thing I like about Don't Starve is that unlike Minecraft and a lot of other sandboxes, everything that has been loaded continues doing stuff, even if you're off the screen. So yes, they will burn to ashes.