Rapidly accreting massive protostars undergo a phase of deuterium shell burning during pre-main-sequence evolution that causes them to swell to tenths of an AU in radius. During this phase, those with close binary companions will overflow their Roche lobes and begin transferring mass. Since massive stars frequently have companions at distances well under 1 AU, this process may affect the early evolution of a substantial fraction of massive stars. We use a simple protostellar evolution model to determine the range in accretion rates, mass ratios, and orbital separations for which mass transfer will occur, and we compute approximately the stability and final outcome of the transfer process. We discuss how mass transfer affects the demographics of massive binaries and show that it provides a natural explanation for the heretofore unexplained population of massive "twins," high-mass binaries with mass ratios very close to unity.