I have an iPhone and use Lightroom CC. But the app takes up all of my storage on my phone with photos. Is there a way to have only a few photos downloaded and the rest just sitting in the cloud to be called up as needed?
Import and share, Performance, Presets or profiles
As Akash said, the setting to "Only Download Smart Previews" will save space on your phone, but if you have not been using this setting so far, and you apply it now, I think it will move the originals into Cache. So you might want to see what the size the Cache is in Local Storage settings then "delete Cache" to get the space back if that happens.
It is currently possible to selectively store originals locally on the phone, but only by album. Click the three dots next to an album and select "Store Locally".