I need to free up space on my iPhone. How do I keep photos on iCloud and remove them from the iPhone?

iCloud Photos is a syncing service. Any photo you delete from a device where iCloud Photos is turned on will also be deleted from iCloud and from any other device connected to iCloud Photos. You can potentially reduce the storage space required on a device by turning on Optimize Storage. See: https://support.apple.com/en-us/HT204264

Apple does not currently offer cloud-only storage for photos. If that is what you want or need, you may want to explore 3rd party options such as Google Photos or Dropbox.

Another way that photos may be on iCloud (if you don’t have iCloud Photos turned on) is via a backup of an iOS device. The only thing you can do with a backup is use it to restore a device or set up a new device. The contents of backups are not viewable at iCloud.com and if you delete a photo from a device it will not be included in future backups and will be lost.