What metabolic activities are performed by a developing human fetus's liver?
It's old and I can't get access to this issue of the Annals of the New York Academy of Science, but it looks like it has some relevant information. Sifting through the abstracts it seems the vasculature of the fetal liver is completed at around 8 weeks although is still different to the adult vasculature because of the umbilical vein. The growth of the organ itself goes through different stages with each lobe growing different amounts:


They start the same size
12 weeks the left lobe is twice as large as the right
36 weeks the left is 10% larger than the right
at birth they are the same size
5days after birth the right is 10% larger
adults, they are the same size


And, studies have been conducted with dyes to look at liver function in both its capacity to concentrate the dyes and secrete it into the bile. The fetal liver can carry out bile acid synthesis, despite lacking digestive purposes, from very early on (Suchy et al. 1987). The fetus does however require help in removing the toxic cholephilic organic anions (COAs), which is done via the placenta. This review describes more.

There is also evidence to suggest the newborn's liver is not performing all the usual metabolic processes of children or adults (from looking at drug toxicity and delays in excretion/detoxification), as would be expected. More recent literature notes the fetal liver lobes (right and left) as having differnt biochemical capacities and independent function due to the right one getting less oxygen. Also, a layer of drug metabolising enzymes normally present in adults is missing in the fetal liver, although its capacity for drug metabolism is still better than that of rodents.


  In the adult liver, there is a clear zonation of metabolism of carbohydrates, proteins, lipids, and bile salts. In the livers of fetal rats, mice, and humans, this heterogeneous distribution of enzymes and metabolic function does not exist initially. Towards the end of gestation, the adult pattern of heterogeneity for these processes begins to emerge, and after birth, the adult pattern is rapidly established (Ring et al. 1999)


There is in vitro evidence for the activity of glutathione S-transferase and sulfotransferases in the human fetal liver. Returning to the asymmetry of fetal liver function (albeit in sheep..) it was found that levels of malondialdehyde, an index of lipid peroxidation, oxygen consumption, and CYP levels (a family of oxidative enzymes), are higher in the left lobe than the right lobe of the fetal liver which evens out after birth.

So in conclusion, looks like a variety of metabolic enzymes are present in the human fetal liver from very early on after the organ is formed. We know a varitey of metabolic enzymes are present and the drug/dye metabolism studies show there is some fetal liver activity, albeit reduced from that of an adult or child. The limitation with many of the studies that have been done is that they have to use in vitro data and extrapolate to the in vivo situation, so I guess we still don't really know for sure how metabolically active it is. It is clear though that it remains inefficient and unable to carry out all the functions required of it during childhood/adulthood until after further development after birth when it has stopped growing and adapted to ex utero conditions. 

References

Ring JA, Ghabrial H, Ching MS, Smallwood RA, Morgan DJ. Fetal hepatic drug elimination. Pharmacol Ther. 1999 Dec;84(3):429-45.

Suchy FJ, Bucuvalas JC, Novak DA. Determinants of bile formation during development: Ontogeny of hepatic bile acid metabolism and transport. Semin Liver Dis 1987; 7(2): 77-84