If your Dropbox folder is full and you have no free storage space, then file syncing will stop. Files won't be removed, and syncing will resume when you bring your account under your storage space quota again.
To get your account under your storage quota you can do any of the following:
How much storage space do I have?
A Dropbox Basic account comes with 2GB of storage space and is always free. Dropbox Basic is best if you want to try out Dropbox, or if you're only using Dropbox for personal files.
If you need more storage space or features, read about the different Dropbox plans and choose the right plan for you.