I've yet to stay concierge, but I would love to & have asked this question before The nearly unanimous answer was NO...folks said they pay lots of extra $$ to stay concierge & no tip is necessary.
The AAA travel books suggests that you would tip everyone in the hotel, maids, turndown maids, bellhops, waiter, room service, valet, except for front desk, maint., and concierge. the books does say to tip the concierge only if you ask them to do something out of the ordinary from there duties or very strange request. Maid and turndown maid should get $2 per night, the bellhop should get 1-2 a bag according to the book.