The effect of nonmagnetic impurities on the phase diagram of the classical Heisenberg antiferromagnet on a triangular lattice is investigated. We present analytical arguments confirmed by numerical calculations that at zero temperature vacancies stabilize a conical state providing an example of "order by quenched disorder" effect. Competition between thermal fluctuations and the site disorder leads to a complicated H-T phase diagram, which is deduced from the classical Monte Carlo simulations for a representative vacancy concentration. For the XY triangular-lattice antiferromagnet with an in-plane external field, nonmagnetic impurities stabilize the fanlike spin structure. We also briefly discuss the effect of quantum fluctuations.