Nope. Of course, you'll see kids AND adults in shorts. But if you are asking what Carnival *asks* you to do, no shorts on anyone at night in the dining room.
My son wore khaki pants and a collared shirt for the dining room, except on formal night when he wore slacks, a button down shirt, tie and navy blazer. Most of the kids in our dining room were dressed up for formal night and we didn't see any shorts on the other nights, although I'm sure there were some.
CarnivalSensation February 2015
Carnival Dream November 2015
Carnival Fascination April 2014
Carnival Elation March 2011
Carnival Imagination Sept 2007
Carnival Sensation Dec 2006
RCI Sovereign of the Seas Sept 2006
Carnival Miracle Sept 2005
Carnival Glory Sept 2004
Carnival Fantasy Jan 2004
My kids wore casual dresses (girls - 11 & 14) or khaki pants and a shirt with a collar (boys -9 & 18)) on non-formal nights and on formal nights the girls wore short dressy dresses and the boys wore suits and ties.
In setting the dress code (or "recommendations", as Carnival calls them), children aren't differentiated from adults. I can't think of any reason the guidelines that apply to adults wouldn't apply to kids...