Based on my past experience in Java I would say it is better to close the connection, it probably doesn't matter in a small Android application, but if you have 10 applications running and all of them access the database, you have 10 pending connections. Start a few more and sooner or later another application will have to wait because the SQL server can't handle any more requests.
I guess you could think of it as a file on your computer. You read data from it, and then close it when your done. Why keep a file open in your application?
Now I'm very new to Android programming so I haven't got around to implement database calls. But when I faced the same problem in a Java application a few years ago I implemented a database object, in which I had the connection to the database. "Everyone else" (the classes) had to call the database object (singleton or final methods) to get data, sort of like stored procedures but in the application instead.
Because of this I knew when the calls where made and when they stopped. I then put in a timeout, so as if nothing happened in a few minutes, I would close the connection to the db. (This also took care of some timeout exceptions because the timeout of the connection would never happen.) When a new call entered, I could easily start a new connection and use the new db connection.
Basically I abstracted away SQL calls by having methods as public Fungus[] getAllFungus()
and public Fungus[] getFilteredFungus(string where)
.