This is an expansion of @Michael Lugo's comment.
The proposition is not true even if all denominators have to be unique. A counterexample: $(2,3,4,5,7,8,9,10,12,14,16,49,35280)$.
Here is a Python program which checks that these numbers are different, and the sum of the reciprocals of these numbers is 2, and that there is no subset of these numbers, where the sum of the reciprocals of the numbers in the subset is 1:
import fractions, itertools
R = fractions.Fraction
D = (2,3,4,5,7,8,9,10,12,14,16,49,35280)
assert len(D) == len(set(D)) # All elements are different.
assert sum(R(1,d) for d in D) == 2 # Sum of reciprocals is 2.
for s in xrange(len(D) + 1):
for e in itertools.combinations(D, s):
assert sum(R(1,d) for d in e) != 1 # Sum of reciprocals in subset is not 1.