Living in the desert, do I have to oil wood that is treated?

I certainly would. There are a number of products on the market that will work. Any home improvement or hardware store will have what you need. Here are some links to get you started.

http://www.doityourself.com/stry/how-to-properly-care-for-outdoor-wood-furniture

https://www.amazon.com/dp/B003OF7AGU/ref=asc_df_B003OF7AGU5208242/?tag=hyprod-20&creative=394997&creativeASIN=B003OF7AGU&linkCode=df0&hvadid=167138874211&hvpos=1o3&hvnetw=g&hvrand=14027684320634013868&hvpone=&hvptwo=&hvqmt=&hvdev=t&hvdvcmdl=&hvlocint=&hvlocphy=9031180&hvtargid=pla-309739157932

https://www.amazon.com/dp/B00M9GEFMW/ref=asc_df_B00M9GEFMW5208198/?tag=hyprod-20&creative=395033&creativeASIN=B00M9GEFMW&linkCode=df0&hvadid=196254346466&hvpos=1o9&hvnetw=g&hvrand=14027684320634013868&hvpone=&hvptwo=&hvqmt=&hvdev=t&hvdvcmdl=&hvlocint=&hvlocphy=9031180&hvtargid=pla-316339826318

Is the wood drying out?

I would definitely agree with Naomie. If you don't oil it, the heat and sun will dry it out and break down the wood.

you may have to refinish the bench as the oil will not adhere to stain.Use a weather resistant stain finish

http://www.woodworkersjournal.com/understanding-oil-wax-finishes/

I bought linseed oil and coated my swing. Gave it life for another 2 yrs. Eventually all the wood cracked and gave out.

You definitely want to treat it. The desert air will dry it out.