We examine the effects of the submesoscale in mediating the response to projected warming of phytoplankton new production and export using idealized biogeochemical tracers in a high-resolution regional model of the Porcupine Abyssal Plain region of the North Atlantic. We quantify submesoscale effects by comparing our control run to an integration in which submesoscale motions have been suppressed using increased viscosity. The warming climate over the 21st century reduces resolved submesoscale activity by a factor of 2-3. Annual new production is slightly reduced by submesoscale motions in a climate representative of the early 21st-century and slightly increased by submesoscale motions in a climate representative of the late 21st-century. Resolving the submesoscale, however, does not strongly impact the projected reduction in annual production under representative warming. Organic carbon export from the surface ocean includes both direct sinking of detritus (the biological gravitational pump) and advective transport mediated pathways; the sinking component is larger than advectively mediated transport by up to an order of magnitude across a wide range of imposed sinking rates. Submesoscales are responsible for most of the advective carbon export, however, which is thus largely reduced by a warming climate. In summary, our results demonstrate that resolving more of the submesoscale has a modest effect on present-day new production, a small effect on simulated reductions in new production under global warming, and a large effect on advectively-mediated export fluxes.