Say I have a whole genome expression data (transcriptome) for some sampled human tissue. Is this data sufficient to find out the continental ancestry group of a donor?
If you can call genomic variants from your data, depending on your coverage of the genome, you should be able to map ancestry. For example, if you have sequencing reads, you can align your reads to a reference genome and then call variants using existing pipelines.
Once you have variant calls, you could use ancestry-informative markers to infer the ancestry of your sample directly.
Or you could use existing software to map the ancestry of your sample in relation to other samples of known ancestry, using a program like ADMIXTURE, STRUCTURE, EIGENSOFT, or just principal components analysis. For reference samples, you could use a publicly available data set like HapMap 3 or the Human Genome Diversity Project.