Well, for example, in Christchurch, New Zealand, where we've had several large earthquakes and nearly 10,000 aftershocks since September 2010, the city has been altered irreparably. Many buildings have been torn down.
We were expecting a windfall with the Rugby World Cup, but all the Christchurch games were cancelled, as were a lot of tourist visits as a result.
While the CBD still has sections closed off, many of the tourist destinations in the region (Lyttleton, Akaroa, the Antarctic Center, Hagley Park, the Museum, Art Gallery, Mt Hutt Ski Field, Kaikoura and more) are still open or have reopened, and would LOVE tourists to come back in droves!
As you point out, visiting the sites of devastation is not the goal, and indeed in the dire parts of Christchurch they ask people not to come and rubber-neck as the vibrations from cars make some damaged houses shudder, for example. However, even those people would be happy to hear tourists are coming back to the city and region - it's a beautiful part of the world.