We study the effects of chiral constituent molecules on the macroscopic shapes attained by lipid bilayer membranes. Such fluid membranes are beautiful examples of statistical ensembles of random shapes~ sometimes coupled to in-plane order. We analyze them with methods of continuum elasticity theory, generalizing the well-known Canham-Helfrich model, and in particular incorporate the effects of thermal fluctuations. The condition that coordinate choice be immaterial greatly constrains the possible forms of the statistical weights in these systems, leading to very few independent couplings and hence physically simple models. Thermal fluctuations effectively reduce the chirality of a membrane at long scales leading to an anomalous scaling relation for the radius of bilayer tubules and helices as a function of chirality. En route to this conclusion we develop a perturbative calculation scheme, paying particular attention to the functional measure needed to describe fluctuations covariantly