After a quick cross-referencing of IBDB and TonyAwards.com it appears that
Legally Blonde (as mentioned above) and
Goodtime Charley in 1975 both had 7 Tony nominations without a nod for Best Musical.
A few others of note:
The original production of
Camelot was nominated for 5 Tonys - but not Best Musical.
The Roar of the Greasepaint... - 6 nominations
Goldilocks - 5 nominations
The Act - 6 nominations
"You pile up enough tomorrows, and you'll find you are left with nothing but a lot of empty yesterdays. I don't know about you, but I'd like to make today worth remembering." --Harold Hill from The Music Man