There has been speculation of a class of relativistic explosions with an initial Lorentz factor smaller than that of classical Gamma-Ray Bursts (GRBs). These dirty fireballs would lack prompt GRB emission but could be pursued via their optical afterglow, appearing as transients that fade overnight. Here we report a search for such transients (transients that fade by 5-$sigma$ in magnitude overnight) in four years of archival photometric data from the intermediate Palomar Transient Factory (iPTF). Our search criteria yielded 45 candidates. Of these, two were afterglows to GRBs that had been found in dedicated follow-up observations to triggers from the Fermi GRB Monitor (GBM). Another (iPTF14yb; Cenko et al. 2015) was a GRB afterglow discovered serendipitously. Two were spurious artifacts of reference image subtraction and one was an asteroid. The remaining 37 candidates have red stellar counterparts in external catalogs. The photometric and spectroscopic properties of the counterparts identify these transients as strong flares from M dwarfs of spectral type M3-M7 at distances of d ~ 0.15-2.1 kpc; two counterparts were already spectroscopically classified as late-type M stars. With iPTF14yb as the only confirmed relativistic outflow discovered independently of a high-energy trigger, we constrain the all-sky rate of transients that peak at m = 18 and fade by $Delta$2 mag in $Delta$3 hr to be 680 per year with a 68% confidence interval of 119-2236 per year. This implies that the rate of visible dirty fireballs is at most comparable to that of the known population of long-duration GRBs.
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