I moved your question to the InDesign forum where I think you'll get better answers.
There is no export to PSD feature like in Illustrator. The only way I've been able to do it is to keep both InDesign and Photoshop open. In Photoshop, choose the Type tool and drag out the area text boundary. Switch to InDesign and copy the text you want. Switch to Photoshop and paste. I takes on the attributes you've set in Photoshop.
I don't think there is any way to copy formatting from InDesign to Photoshop. There are too many differences in the text engine in InDesign and in Photoshop. Photoshop's text engine is similar to that in Illustrator.
My reply in the PS forum:
Is there a reason for doing this? The graphics created in Photoshop for slicing are a little better for the web (more effects, better save for web functionality etc) but you can actually create a reasonably efficient web site in InDesign - as long as you map all the styles and then export as HTML.
However, to answer your question, yes and no. You can copy the text box from InDesign and paste in Photoshop which will place it as a Vector Smart Object. It will preserve most, if not all of the styling you have applied. Then to edit this, you simply right click the layer and edit in illustrator. If you copy only the words, it will not paste like this, you need to select the text box.
So, I think those are your two options. You can keep the styling of the InDesign type using Rik's method, but you have to jump into Illustrator to do editing.
Or, you can apply the styling in Photoshop and edit the type there, if you copy just the text, but not the frame.