Yeast can make proteins from urea and sugar. It would be very useful for humans to do the same thing as it would enable us to make use of abundant but low-protein foods. Why can't we, or other animals, do this?

you mean yeast can synthesise all 20 amino-acids where as higher organisms can't - hence the term "essential amino acids". for possible reasons see:
http://www.ncbi.nlm.nih.gov/pubmed/1646 … t=Abstract