  According to the no-hair theorem, astrophysical black holes are fully
characterized by their masses and spins and are described by the Kerr metric.
This theorem can be tested observationally by measuring (at least) three
different multipole moments of the spacetimes of black holes. In this paper, we
calculate the profiles of fluorescent iron lines emitted from the accretion
flows around black hole candidates within a framework that allows us to perform
the calculation as a function of its mass and spin as well as of a free
parameter that measures potential deviations from the Kerr metric. We show that
such deviations lead to line profiles that are significantly altered and may
exhibit a modified flux ratio of the two peaks in their characteristic
double-peaked shape. We also show that the disk inclination can be measured
independently of the spin and the deviation parameter at low to intermediate
inclination angles as in the case of Kerr black holes. We estimate the
precision that near-future X-ray missions such as Astro-H and ATHENA+ are
required to achieve in order to resolve deviations from the Kerr metric in iron
line profiles and show that constraints on such deviations will be strongest
for rapidly spinning black holes. More generally, we show that measuring the
line profile with a precision of ~5% at disk inclinations of 30 or 60 degrees
constraints the deviation parameter to order unity for values of the spin
a>0.5M.
