We present results on the dust attenuation curve of z~2 galaxies using early observations from the MOSFIRE Deep Evolution Field (MOSDEF) survey. Our sample consists of 224 star-forming galaxies with nebular spectroscopic redshifts in the range z= 1.36-2.59 and high S/N measurements of, or upper limits on, the H-alpha and H-beta emission lines obtained with Keck/MOSFIRE. We construct composite SEDs of galaxies in bins of specific SFR and Balmer optical depth in order to directly constrain the dust attenuation curve from the UV through near-IR for typical star-forming galaxies at high redshift. Our results imply an attenuation curve that is very similar to the SMC extinction curve at wavelengths redward of 2500 Angstroms. At shorter wavelengths, the shape of the curve is identical to that of the Calzetti relation, but with a lower normalization (R_V). Hence, the new attenuation curve results in SFRs that are ~20% lower, and log stellar masses that are 0.16 dex lower, than those obtained with the Calzetti attenuation curve. Moreover, we find that the difference in the reddening---and the total attenuation---of the ionized gas and stellar continuum correlates strongly with SFR, such that for dust-corrected SFRs larger than 20 Msun/yr assuming a Chabrier IMF, the nebular emission lines suffer an increasing degree of obscuration relative to the continuum. A simple model that can account for these trends is one in which the UV through optical stellar continuum is dominated by a population of less reddened stars, while the nebular line and bolometric luminosities become increasingly dominated by dustier stellar populations for galaxies with large SFRs, as a result of the increased dust enrichment that accompanies such galaxies. Consequently, UV- and SED-based SFRs may underestimate the total SFR at even modest levels of ~20 Msun/yr. [Abridged]
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