  We consider the potential for the Transiting Exoplanet Survey Satellite
(TESS) to detect transit timing variations (TTVs) during both its nominal and
extended mission phases. Building on previous estimates of the overall yield of
planetary systems from the TESS mission, we predict that during its nominal
two-year mission, TESS will observe measurable TTVs in $\sim30$ systems, from
which $\mathcal{O}(10)$ planet will get precise mass measurements from TTVs
alone, $\sim 5$ planets will have significant constraints placed on their
masses from TTVs, and over a dozen systems will be singly-transiting TTV
systems. We consider a number of different extended mission scenarios, and
predict that in a typical scenario, an extended mission will allow TESS to
increase the number of systems with measurable TTVs to a total of $\sim90$,
from which $\sim 15$ planets will have precise mass measurements, another $\sim
15$ will have significant constraints placed on their masses, and $\sim60$ will
be singly-transiting TTV systems. We also describe how follow-up transit
observations of multi-planet systems discovered by the TESS mission can be
optimally planned to maximize TTV mass and eccentricity constraints.
