Truth is your performance and load testing will often find most of these kinds of issues. Now it won't tell you WHAT is causing the problem but it will tell you THAT it's happening in a reproducible way.
I agree with Peter though - it's not test's job to fix them. Finding that they are happening does help development fix it though.
As said by Wholly , the problem of memory leaks can be detected only in Performance testing.
I feel doing stress testing (one of the testing while checking the performance of application) for that application will solve the problem. The basic goal of stress testing is to find out memory leaks as well as data loss. To do this, you take critical features of application and simulate traffic for ‘n’ number of times in a load testing tool.
Make sure that you do the testing for un interval loads and peak loads to check the stress point of the application. As usual, it depends on the application and user requirements, what are the customer expected loads and peaks.