As detailed in Apple's support article, you can disable iCloud Photo Library on your iOS device by tapping Settings > iCloud, and turning off "Photos".
Your photos will still remain in iCloud Photo Library but will be removed from your device.
My understanding is that iCloud is for storage of photos, and not just for synchronization; otherwise, why would you have the option to purchase more space (which I have done). That being said, @Jaime Santa Cruz is correct in that you cannot back up Time Machine to iCloud, but you should be able to indefinitely store photos, docs, notes, etc., so long as you keep within the storage limit or purchase (and continue to pay) for additional storage.
I had the same problem and actually wrote a question about it just the other day and received a number or comments and an answer. Deleting Photos, what iCloud saves and how to access things from iCloud and device storage issues It seemed clear from what others were saying that I should be able to see icons in iCloud, such as photos, and when you click on/open these tabs you should be able to see your photos/docs. I could not.
It was suggested I contact Apple support if there was nothing there aside from Find My Phone (which there was not - sounds like this is what you are saying). I did contact Apple and sure enough, there was a problem on iCloud/Apple's end and they need to remotely reset my iCloud account(this is the word they used, from a technical perspective I have no clue what was wrong or what they did to fix it), but after about 15 minutes with them iCloud appeared as it should and never had before. There was an orange background with most of the app icons that appear on the dock, and I could now view my photographs and documents, etc.
If iCloud is turned on, your devices are synced and you are going to iCloud.com and getting a black background with just a FMP, you probably need to do what I did.
Best Answer
The only way to remove the photos from your Mac is to sign-out of iCloud. This will keep the photos in iCloud but remove them, and all other iCloud-synced data, from the Mac.
iCloud is a syncing solution not an offline storage solution. It's designed to give you access to your data from any device that logged-into that iCloud account.
From Apple's iCloud page:
You could use an external storage device to store your photos and that will free-up your Air's HDD space.