We discuss transport through an interferometer formed by helical edge states tunnel-coupled to metallic leads. We focus on the experimentally relevant case of relatively high temperature as compared to the level spacing and discuss a response of the setup to the external magnetic flux φ (measured in units of flux quantum) piercing the area encompassed by the edge states. We demonstrate that tunneling conductance of the interferometer is structureless in the ballistic case but shows sharp antiresonances, as a function of magnetic flux φ - with the period 1/2 - in the presence of a magnetic impurity. We interpret the resonance behavior as a coherent enhancement of backward scattering off the magnetic impurity at integer and half-integer values of flux, which is accompanied by suppression of the effective scattering at other values of flux. Both enhancement and suppression are due to the interference of processes with multiple returns to the magnetic impurity after a number of clockwise and counterclockwise revolutions around the setup. This phenomenon is similar to the well-known weak-localization-induced enhancement of backscattering in disordered systems. The quantum correction to the tunneling conductance is shown to be proportional to the flux-dependent "ballistic Cooperon." The obtained results can be used for flux-tunable control of the magnetic disorder in Aharonov-Bohm interferometers built on helical edge states.