But a better solution would be to use something like the Spring Framework that wraps that stuff automatically. Less coding, less debugging, less chance someone will forget to do proper cleanup.
Incidentally, apps I've seen that were prone to leak memory very commonly did so precisely because they didn't have proper interception and cleanup of errors. Error-based leaks are slower to accumulate than most others, but sooner or later, they will add up.
And yest, you should close connections. If they're pooled connections and you don't close them, you won't only leak memory, you'll leak connections, and thereby eventually exhaust the connection pool. Which will bring you down a lot faster than memory leaks.
Customer surveys are for companies who didn't pay proper attention to begin with.