The connection between the production of the cosmogenic isotope 10Be and changes in heliomagnetic activity makes ice core 10Be an attractive proxy for studying changes in solar output. However, interpreting 10Be ice core records on centennial timescales is complicated by potential climate-related deposition changes that could obscure the 10Be production signal. By using the Goddard Institute for Space Studies ModelE general circulation model to selectively vary climate and production functions, we model 10Be flux at key ice-coring sites. We vary geomagnetic field strength and the solar activity modulation parameter ($phi$), CO2, sea surface temperatures, and volcanic aerosols to assess impacts on 10Be. Specifically, we find significant latitudinal differences in the response of 10Be fluxes to changes in the production function. In the climate experiments the 10Be deposition changes simulated over ice sheets in both hemispheres are comparable to those seen in the production experiments. This altered deposition combined with changes of snow accumulation results in significant climate-related 10Be concentration variation in both Greenland and Antarctica. Over the Holocene our results suggest that the 10Be response to climate change should not be neglected when inferring production changes.