In this paper we describe a uniform analysis of eight transits and eleven secondary eclipses of the extrasolar planet GJ 436b obtained in the 3.6, 4.5, and 8.0 micron bands using the IRAC instrument on the Spitzer Space Telescope between UT 2007 June 29 and UT 2009 Feb 4. We find that the best-fit transit depths for visits in the same bandpass can vary by as much as 8% of the total (4.7 sigma significance) from one epoch to the next. Although we cannot entirely rule out residual detector effects or a time-varying, high-altitude cloud layer in the planets atmosphere as the cause of these variations, we consider the occultation of active regions on the star in a subset of the transit observations to be the most likely explanation. We reconcile the presence of magnetically active regions with the lack of significant visible or infrared flux variations from the star by proposing that the stars spin axis is tilted with respect to our line of sight, and that the planets orbit is therefore likely to be misaligned. These observations serve to illustrate the challenges associated with transmission spectroscopy of planets orbiting late-type stars; we expect that other systems, such as GJ 1214, may display comparably variable transit depths. Our measured 8 micron secondary eclipse depths are consistent with a constant value, and we place a 1 sigma upper limit of 17% on changes in the planets dayside flux in this band. Averaging over the eleven visits gives us an improved estimate of 0.0452% +/- 0.0027% for the secondary eclipse depth. We combine timing information from our observations with previously published data to produce a refined orbital ephemeris, and determine that the best-fit transit and eclipse times are consistent with a constant orbital period. [ABRIDGED]
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