  Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity
conditions for vector fields on the Riemannian warped product $B\times_fF$,
with $f:B \longrightarrow ]0,+\infty[$, using a characteristic variational
condition. Then, we apply this to the case $B=\mathbb{R}$ and $F$ is a
three-dimensional connected Riemannian Lie group $G$ equipped with a
left-invariant metric, to determine harmonic vector fields on
$\mathbb{R}\times_fG$. We give examples of harmonic vector fields on $G$ which
are not left-invariant and determine harmonic vector fields on
$\mathbb{R}\times_fG$. We conclude with some examples of vector fields on
$\mathbb{R}\times_fG$ which are harmonic maps.
