First, a question:
Are you running your project using the commandline of TestComplete or TestExecute? If so, you can turn on a "/silentmode" commandline switch which will automatically close any dialogues that your automation project encounters.
In either case, though, you can adjust how a project runs in playback for errors. In your Project Properties, there's an option for Playback. In there, there's a checkbox for "Error dialog". If that is turned off, when an error occurs, TC will no longer display the dialog but will simply log the error to the TestLog and proceed accordingly.