The sensitivity of the climate system to changes in radiative forcing is crucial for our understanding of past and future climates. Especially important are feedbacks related to melting of ice sheets like the Greenland ice sheet (GIS) and its potential impact on the Atlantic meridional overturning circulation (AMOC). These effects are likely to delay and dampen predicted long-term warming trends. Estimates of climate sensitivity may be deduced from palaeoclimate-reconstructions, but this raises the question whether past climate sensitivity is applicable to the future. Therefore we have analysed the impact of GIS melt water on the AMOC strength in two past warm climates (last interglacial and early present interglacial) and three future scenarios with three different model parameter sets. These model parameter sets represent three different model sensitivities to freshwater perturbation: low, moderate and high. In both the moderate and high sensitivity versions, we find for lower GIS melt rates (below 54 mSv, Sv = 106 m3/s) a clear difference between past and future warm climates in the sensitivity of the AMOC to GIS melt. This difference is connected to the convective activity in the Labrador Sea and the amount of additional surface freshening due to sea ice melting. In contrast, for higher GIS melt rates (over 54 mSv) we find similar reductions of the AMOC strength in all cases. Considering the low sensitivity version of our model, we find that for all GIS melt rates the influence of freshwater forcing on the AMOC is independent of the background climate. Our results and implications are thus strongly determined by the parameter set considered in our model. Nonetheless, our results from two out of three model versions suggest that proxy-based reconstructions of past AMOC sensitivity to GIS melt are likely to be misleading if interpreted for future applications.