SQLite, open one permanent connection or not?

As with all things database, it depends. In this specific case of sqlite, there are two "depends" you need to look at:

1. Are you the only user of the database?
2. When are implicit transactions committed?

For the first item, you probably want to open/close different connections frequently if there are other users of the database or if it's all possible that more than process will be hitting your sqlite database file at the same time.

For the second item, I'm not sure how sqlite specifically behaves. Some database engines don't commit implicit transactions until the connection is closed. If this is the case for sqlite, you probably want to be closing your connection a little more often.

The idea that connections should be short-lived in .Net applies mainly to Microsoft Sql Server, because the .Net provider for Sql Server is also able to take advantage of a feature known as connection pooling. Outside of Sql Server this advice is not entirely without merit, but it's not as much of a given.

If it is a local application being used by only one user I think it is fine to keep one connection opened for the life of the application.

I think with most databases the "Best used and closed" idea comes from the perspective of saving memory by ensuring you only have the minimum number of connections need open.

In reality opening the connection can be a large amount of of overhead and should be done when needed. This is why managed server infrastructure (weblogic etc.) promotes the use of connection pooling. In this way you have N connections that are utilizable at any given time. You never "waste" resources but you also aren't left with the responsibility of managing them at a global level.