I tried to leave a connection open once -- I used it to stuff a repeater or something -- can't remember now.
Later in the program, I had another need to use the connection -- I think I had it so that when a user clicked a parent item in the repeater, a detail div would pop-up with more information for that item. This generated an error -- something to the effect of 'cannot open() on an open connection'.
I think the error might have been avoidable another way (like checking to see if the connection I was trying to open was already open), but as I thought about it, I realized I'd have to make that a standard practice throughout my app, and that seemed like too much work, so I just made it a standard practice to always close my connections after each use.
Connections stay in a connection pool -- I'm no whiz on that -- but if was curious about performance, I guess I'd keep that in mind, in terms of what it costs to open a connection multiple times -- whatever your situation is requiring anyway.
Another thought is that your DB admin may be able to force-close all open connections, or the db may close for some other reason. If you're not the dba, you might give a thought to risk/benefit of depending on something that you don't have long-term control over like keeping the connection open.