  We study the variation of the effective fine structure constant alpha for
Dirac-Born-Infeld (DBI) type dark energy models. The DBI action based on string
theory naturally gives rise to a coupling between gauge fields and a scalar
field responsible for accelerated expansion of the universe. This leads to the
change of alpha due to a dynamical evolution of the scalar field, which can be
compatible with the recently observed cosmological data around the redshift
$\tilde{z} \lesssim 3$. We place constraints on several different DBI models
including exponential, inverse power-law and rolling massive scalar potentials.
We find that these models can satisfy the varying alpha constraint provided
that mass scales of the potentials are fine-tuned. When we adopt the mass
scales which are motivated by string theory, both exponential and inverse
power-law potentials give unacceptably large change of alpha, thus ruled out
from observations. On the other hand the rolling massive scalar potential is
compatible with the observationally allowed variation of alpha. Therefore the
information of varying alpha provides a powerful way to distinguish between a
number of string-inspired DBI dark energy models.
