Zero Install is designed to be largely maintenance-free, e.g., checking for updates automatically. However, if you want to manually ensure everything is running optimally you can use the following commands:

Download and install updates for Zero Install itself if available:
0install self-update --batch

Download and install updates for applications integrated by Zero Install and remove any outdated files:
0install update-all --clean --batch

Find and merge any duplicate files in the cache:
0install store optimise --batch