Can I leave a suspension fork upside down? I currently have my mountain bike disassembled, I broke a tire and needed some space in my room. The frame is resting upside down on the saddle and handlebar.

The fork itself is a Rock Shox SID Dual Air (pretty old model, I guess 2001). As this fork relies - to my best knowledge - only on air, storing it upside down shouldn't be a problem. Am I correct, what about other types of suspension forks?
You will not damage a suspension fork by hanging it upside down. Although the fork is air sprung it also uses oil for dampening in one of the legs. Oil may leak if your seals are degraded. The seals in suspension forks are made to ensure they don't react with the suspension oil. The oil will actually help lubricate the seals.

This is the same for coil (spring based) forks.

Once you refit your fork it will take time to settle back to its original state.