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I have two DEMs of an open-pit mine taken a few weeks apart. I would like to estimate the volume that has been excavated during that interval, ie the difference of volume between the two DEMs.

Can I do that with GRASS? If so how?

If not, do you have any suggestion on how to do it? (I would prefer using an open source software...)

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The difference of two rasters, before and after, can be produced as a new raster with r.mapcalc (rasterDiff= rasterBefore - rasterAfter).

Then, you can have the volume by using the r.report of rasterDiff and multiplying the results by the cell size.

You can go to step 6 of this Q&A How to restore a scarp surface grid to calculate material loss

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