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I have a shapefile that is composed of multiple overlapping polygons that represent extent of habitats of certain species. Here is a 1 MB example of such file from International Union for the Conservation of Nature website. I would like to convert the shapefile into multiple GeoTIFFs such that each file contains range map of a single species. How can I do that?

In ArcGIS Info I tried "polygon to raster", but the resulting raster is composed of a single band and multiple values (so that pixel values range from 0 to N, where N is number of species). This way information about overlap is lost. If I could convert my shapefile to multiband (such that each band represents a single species), the perhaps I could use the solution from this post. It would be best if my output images had 1 bit per pixel, since the range map should be essentially composed of zeros (species not present) and ones (species present).

Unless you know better options than ArcGIS? I also have at my disposal QGIS, GRASS and command-line GDAL tools.

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You need to select the polygons sharing a particular species before running polygon to raster, and have to repeat the operation for each species. Geoprocessing tools should honor selections for inputs, so the resulting raster will only be for one species. Depending on how many you have and how often this might be repeated, you might want to consider building a model or scripting the process with an iterator on the species attribute.

Alternatively you could split the shapefile into a different shapefile for each species using a split by attributes tool (mentioned at other questions here) and process them one at a time.

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