I wasn't sure if challenge xp was included or the xp just from getting objectives and kills count in double xp.
I believe its double ALL XP earned.
I haven't seen anything to the contrary.
I don't remember what it was for Black Ops, but for MW3, Challenge XP was definitely not doubled.
I remember it coming up in the MW3 forums where people were thinking about stopping just shy of certain challenges that would give them 10,000XP so they could double it, but it didn't double.
Challenges for weapons do...
... I saved getting 150 kills with no perks/attachments for a prestige to unlock Gold and Diamond. In one game...
... I got 69k XP.
I seem to remember it does. Or at least it did last double XP weekend. Might be just the weapons like the guys say above. Seems like ages ago.