The obvious first-order approach is to simply set the time of the system in question to a date past 2038 and see if there are issues. In some cases, this kind of ‘black box’ testing may work, but there are a lot of things to consider. Depending on the system, you may not be able to change the date on a live system without repercussions, so you may need to set up a sandbox in which to test. Depending on the size and complexity of the system, the time and cost to do this may be significant, but there may be no other way.
There are some commercial products available for testing future dates on Linux, Windows, and z/OS.