If the text box of King Robert's Chambers (or any of the others) that was played as an attachment gets blanked, does it get discarded?

Yes. The rules say that unless otherwise specified by card text, attachments can only be attached to characters. Without the text actually allowing you to attach the Chambers to the House card, it thus becomes illegal (since it is not attached to a character). Illegal attachments are immediately discarded.

I wasn't clear on the situation because of cards like Abandoned Forge that attach cards to non-character cards through effects other than their own text.

Khudzlin said:

Thanks for the clarification.