What happens if I remove my dropbox account from my phone so I can add another account? I want to add another account but I have to remove the other one first. Will that delete all my photos? Or can I still access my photos if I log in on my dropbox account online?
Logging out will not delete your photos or other content. You will still be able to access all content in the old account via re-logging in on that device, another device, or in browser.

However if you have multiple accounts, you may want to be sure that you are using each account in accordance with the terms and conditions of service of Dropbox.