Correctly open/close a database with Singleton design pattern
You should call close anytime you are done writing to your database. For example when you insert data, you will have an open connection to the database that should be closed when it is done.
Reading is different. When you create a SQLite database on your phone, the data is persistent. The database exists and the handler you create provides a convenient way to access that information. Reading the database usually takes place by getting a readable instance of the database and using a Cursor to extract values. In that case you close the cursor when you're done, not the database itself.
You're right that you should not be closing the database connection during separate activities' lifecycle methods. Instead, as suggested above, close the database connection in your handler's methods that write to the database when you are done performing that transaction.