A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal G-bundle is a Lie subgroup of G (Cartan had earlier asserted this, but apparently without proof).
This restriction, that it be a Lie subgroup, allows for a lot of poorly-behaved subgroups, for example a line with irrational slope on a torus. This subgroup comes from a perfectly fine immersion of the Lie group ℝ, but it's not closed in the induced topology of the torus.
As an example of something that's not a Lie subgroup, let G= ℝ, consider an uncountable set of ℚ-independent points, none of which are rational, and consider the subgroup they generate. If this were a Lie subgroup it would be the image of an uncountable discrete space (there can't be anything 1-dimensional, since we left out the rationals), which wouldn't be second countable, hence not a manifold and not a Lie group.
This seems like a pretty contrived example, and I suspect there is more content to 'being a Lie subgroup' than having countably many components. However, I can't seem to pin down something that would illustrate this. Can anyone give me an example of a connected subgroup of a Lie group that is not a Lie subgroup?