Here is an article with a short proof of Jones' theorem on existence of dense connected proper subgroups of $\mathbb R^2$:
Maehara, Ryuji, On a connected dense proper subgroup of $\mathbb R^2$ whose complement is connected, Proc. Am. Math. Soc. 97, 556-558 (1986). ZBL0593.54037.
On the other hand, a path-connected subgroup of a Lie group is always a Lie subgroup:
Yamabe, Hidehiko, On an arcwise connected subgroup of a Lie group, Osaka Math. J. 2, 13-14 (1950). ZBL0039.02101.
hence, there are no path-connected dense proper subgroups in $\mathbb R^n$.