  We consider a quantum particle subject to Ohmic dissipation, moving in a
bichromatic quasiperiodic potential. In a periodic potential the particle
undergoes a zero-temperature localization-delocalization transition as
dissipation strength is decreased. We show that the delocalized phase is absent
in the quasiperiodic case, even when the deviation from periodicity is
infinitesimal. Using the renormalization group, we determine how the effective
localization length depends on the dissipation. We show that {a similar problem
can emerge in} the strong-coupling limit of a mobile impurity moving in a
periodic lattice and immersed in a one-dimensional quantum gas.
