Much of the science from the exoplanets detected by the TESS mission relies on precisely predicted transit times that are needed for many follow-up characterization studies. We investigate ephemeris deterioration for simulated TESS planets and find that the ephemerides of 81% of those will have expired (i.e. 1$sigma$ mid-transit time uncertainties greater than 30 minutes) one year after their TESS observations. We verify these results using a sample of TESS planet candidates as well. In particular, of the simulated planets that would be recommended as JWST targets by Kempton et al. (2018), $sim$80% will have mid-transit time uncertainties $>$ 30 minutes by the earliest time JWST would observe them. This rapid deterioration is driven primarily by the relatively short time baseline of TESS observations. We describe strategies for maintaining TESS ephemerides fresh through follow-up transit observations. We find that the longer the baseline between the TESS and the follow-up observations, the longer the ephemerides stay fresh, and that 51% of simulated primary mission TESS planets will require space-based observations. The recently-approved extension to the TESS mission will rescue the ephemerides of most (though not all) primary mission planets, but the benefits of these new observations can only be reaped two years after the primary mission observations. Moreover, the ephemerides of most primary mission TESS planets (as well as those newly discovered during the extended mission) will again have expired by the time future facilities such as the ELTs, Ariel and the possible LUVOIR/OST missions come online, unless maintenance follow-up observations are obtained.
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