A locally compact example (actually compact abelian) is the Pontryagin dual $G$ of the discrete abelian group $\mathbf{Z}^\omega$.

Indeed

1. a compact abelian group is connected iff its Pontryagin dual is torsion-free;
2. a connected compact abelian group is locally connected iff all subgroups of finite rank of its Pontryagin dual are free abelian (and indeed all countable subgroups of $\mathbf{Z}^\omega$ are free abelian);
3. a compact abelian group $D$ is path-connected iff its Pontryagin dual $D$ satisfies $\mathrm{Ext}^1_\mathbf{Z}(D,\mathbf{Z})=0$ (and $\mathbf{Z}^\omega$ does not satisfy this property).

For all this, see J. Dixmier, Quelques propriétés des groupes abéliens localement compacts, Bull. Sci. Math. (2) 81 1957 38-48. ((1) is immediate, and Dixmier also attributes (2) to Pontryagin.)

Moreover he checked that for second-countable connected compact abelian groups, locally connected $\Leftrightarrow$ path-connected $\Leftrightarrow$ torus (i.e. isomorphic to $(\mathbf{R}/\mathbf{Z})^k$ for some $k\le\omega$), so no example can be found among second-countable locally compact abelian groups.

A locally compact example (actually compact abelian) is the Pontryagin dual $G$ of the discrete abelian group $\mathbf{Z}^\omega$.

Indeed

1. a compact abelian group is connected iff its Pontryagin dual is torsion-free;
2. a connected compact abelian group is locally connected iff all subgroups of finite $\mathbf{Q}$-rank ($\ast$) of its Pontryagin dual are free abelian (and indeed all countable subgroups of $\mathbf{Z}^\omega$ are free abelian);
3. a compact abelian group is path-connected iff its Pontryagin dual $D$ satisfies $\mathrm{Ext}^1_\mathbf{Z}(D,\mathbf{Z})=0$ (and $\mathbf{Z}^\omega$ does not satisfy this property).

($\ast$) a torsion-free abelian group $\Lambda$ has finite $\mathbf{Q}$-rank, often written "finite rank" if $\Lambda\otimes_\mathbf{Z}\mathbf{Q}$ has finite dimension; this means that $\Lambda$ is isomorphic to a subgroup of $\mathbf{Q}^d$ for some finite $d$.

For all this, see J. Dixmier, Quelques propriétés des groupes abéliens localement compacts, Bull. Sci. Math. (2) 81 1957 38-48. ((1) is immediate, and Dixmier also attributes (2) to Pontryagin.)

Moreover he checked that for second-countable connected compact abelian groups, locally connected $\Leftrightarrow$ path-connected $\Leftrightarrow$ torus (i.e. isomorphic to $(\mathbf{R}/\mathbf{Z})^k$ for some $k\le\omega$), so no example can be found among second-countable locally compact abelian groups.