Moving photos from desktop to Dropbox

You can't really remove photos to Dropbox, as the way Dropbox works, a copy is maintained on your hard disk. You also can't store photos in the cloud, because Lightroom requires photos to be stored on a local or network disk, and not in the cloud.

dj is as usual completely correct.

Note that ALL cloud solutions work this way. They maintain a local copy on your hard disk so that you can access the files locally and then it syncs the files that are changed to the cloud servers in the background when needed. This is done to work around the very large latency that you get when going to the internet. So your standard cloud solutions are simply not going to work to release storage on your computer. Some cloud services will allow you to not sync certain folders but then you don't have access to the images in Lightroom! A solution could be to treat a drive on an internet sever as a network drive. This can be done through things like WebDAV on amazon web services but they are in practice not workable solutions for this problem as it is going to be incredibly slow on most broadband connections. You'd need gigabit fiber to make it even remotely workable. This is really just a complex and slow way to implement a NAS.

Only real solutions to running out of storage are to install a bigger internal drive, to use external storage on a fast interface (USB-3, USB-c, thunderbolt or similar) or to use local networked storage (if you have a fast network). Cloud storage doesn't help.