  We study the finite temperature phase diagram of the Heisenberg-Kitaev model
on a three dimensional hyperhoneycomb lattice. Using semiclassical analysis and
classical Monte-Carlo simulations, we investigate quantum and thermal
order-by-disorder, as well as the magnetic ordering temperature. We find the
parameter regime where quantum and thermal fluctuations favor different
magnetic orders, which leads to an additional finite temperature phase
transition within the ordered phase. This transition, however, occurs at a
relatively low temperature and the entropic effects may dominate most of the
finite temperature region below the ordering temperature. In addition, we
explore the magnetization process in the presence of a magnetic field and
discover spin-flop transitions which are sensitive to the applied-field
direction. We discuss implications of our results for future experiments on a
hyperhoneycomb lattice system such as the recently discovered \beta-Li2IrO3.
