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Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb R^k/\mathbb Z^k$.

I know a couple of sources of proofs of the two-dimensional case, which is not so hard, and it seems to me that the classification of subgroups of $\mathbb R^k$ gets me close to this, but does someone have a crisp proof of this fact, or a good reference?

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The result you want goes by the name "Kronecker's theorem". There is a proof in Siegel's "Lecture on the Geometry of Numbers" and in Bump's book on Lie groups.

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