something like:
    test1 <- extract(dryas,teow,na.rm=TRUE,sp=TRUE,fun=function(x) sum(x>0))
might give you what you ask: the area (count of raster cells, so in units
of cell areas) in each polygon of teow with raster values > 0.
If dryas has small cells, relative to the size of the ecoregion polygons in
teow, that should be close enough.  If not, look at the documentation for
raster::extract and use weights=TRUE and a 2-step process (generate
weights, then compute weighted sum by polygon).

I hope that this points you in a useful direction...

Tom

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