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. (In unrestricted access at AMS site)

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 $\mathbb R^n$.

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. (In unrestricted access at AMS site)

On the other hand, a path-connected subgroup of a Lie group is always a Lie subgroup (not necessarily closed):

Yamabe, Hidehiko, On an arcwise connected subgroup of a Lie group, Osaka Math. J. 2, 13-14 (1950). ZBL0039.02101.

On the other hand, every Lie subgroup of $\mathbb R^n$ is closed (since the exponential map is a diffeomorphism); hence, every path-connected subgroup in $\mathbb R^n$ is closed.

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 $\mathbb R^n$.

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. (In unrestricted access at AMS site)

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 $\mathbb R^n$.