There is always a (unique)normal condition expectation onto a masa in a type II_1 factor. When does a masa in a type III factor admit a normal conditional expectation? (If we drop normality, conditional expectations always exist because abelian subalgebras are injective Banach spaces).
Takesaki showed in section 6 of:
MR0303307 (46 #2445) Takesaki, Masamichi Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306–321. http://www.sciencedirect.com/science/article/pii/0022123672900043
that the following are equivalent for a von Neumann algebra M (not necessarily a factor):
Edit: As Jon Bannon helpfully points out, the original question asked "when does a MASA admit a conditional expectation onto it", and so this answer only says "not always" which isn't really a full answer!