I have a test suite that runs a Windows executable with a variety of command line options to test that the executable works correctly. In some test cases, those arguments cause the executable to break badly and this dialog appears:

Unfortunately, this dialog remains until I manually press Abort, Retry, or Ignore. This prevents me from running these tests in an automated fashion. I have tried calling the executable through the Windows CDB Debugger with a variety of CDB options with no luck.
How can I prevent these dialogs from appearing or automatically continue on with the tests?
It is clear from the window title that you are testing the Debug build of the program. This is not a good practice, you are not testing the program the way it is going to run on your customer’s machine. A Debug build is suitable for the programmer to debug their code. But it does occasionally hide a bug. Such bugs can be very hard to diagnose, you need tests to flush them out. You are not performing such a test.
Only test the Release build. Which then also automatically solves this dialog problem, the Release version of the CRT doesn’t display it.
Btw: test failed.