Yes - Some optional features run in the background.
With PDF Viewer for Android, we take great care to only perform absolute necessary tasks to preserve your battery.
If we detect that your device is charging, our Full Text Search indexing process will run and parse your documents for faster searching. it will continue running for a short while after you close the app. This can be disabled by turning off the Full Text Search in the settings.
If you change or upload files inside a Dropbox connection the app will perform the upload even when it is closed to make sure your files are always in sync.