Please keep in mind that I'm a noob, and someone else may come along with better advice.
That said, I had this happen last week with the my TV Guide app, and after searching around it seems that the app crashes from time to time. You might want to look for troubleshooting tips on the App's page in the ITunes store, or go to the NY Times website and search for troubleshooting tips for their app. Or you can trying googling something like "Trouble with NY Times iPhone App" and see what comes up!