  Two major challenges of contemporary cosmology are the Hubble tension and the
cosmic dipole tension. At the crossroad of these, we investigate the impact of
peculiar velocities on estimations of the Hubble constant from time-delay
cosmography. We quantify the bias on the inference of the Hubble constant due
to peculiar velocities of the lens, the source and of the observer. The former
two, which may cancel from one system to another, affect the determination of
the angular diameter distances in the time-delay formula, and reconstructed
quantities like the angle to the source, via a lens model. On the other hand,
the peculiar velocity of the observer, which is a debated quantity in the
context of the cosmic dipole tension, systematically affects observed angles
through aberration, redshifts, angular diameter distance and reconstructed
quantities. We compute in detail the effect of these peculiar velocities on the
inference of the Hubble constant to linear order in the peculiar velocities for
the seven lenses of the H0LiCOW/TDCOSMO collaboration. The bias generated by
the observer's peculiar velocity alone can reach $1.15\%$ for the lenses which
are well aligned with it. This results in a $0.25 \%$ bias for the seven
combined lenses. Assuming a typical peculiar velocity of $300$ km s$^{-1}$ for
the lens and the source galaxies, these add an additional random uncertainty,
which can reach $1\%$ for an individual lens, but reduces to $0.24\%$ for the
full TDCOSMO sample. The picture may change if peculiar velocities turn out to
be larger than expected. Any time-delay cosmography program which aims for
percent precision on the Hubble constant may need to take this source of
systematic bias into account. This is especially so for future ground-based
surveys which cover a fraction of the celestial sphere that is well aligned
with the observer's peculiar velocity.
