Limitations/breakdown of Mermin-Wagner Theorem

As @NorbertSchuch comments, a theorem cannot have counterexamples. Well, at least this is true for what mathematicians call theorems. I thus take the question as asking for a way to violate the "physicists' version of Mermin-Wagner theorem", which would state something like "a continuous symmetry cannot be spontaneously broken in dimensions $1$ and $2$ at positive temperature". In this form (which is the form you often see this result stated in the physics literature), there are of course counterexamples and the latter can be found by trying to remove some of the assumptions of the mathematically precise versions of this theorem.

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Probably the simplest way to violate the (physicists' version of) Mermin-Wagner theorem is to consider a system with sufficiently long-range interactions. For instance, consider the (classical) XY model on $\mathbb{Z}^2$ with Hamiltonian $$ H=-\sum_{i\neq j} J_{|j-i|} S_i\cdot S_j, $$ where $J_r = r^{-\alpha}$. Then, for any $\alpha\geq 4$, the Mermin-Wagner theorem applies (see, for instance, this paper), but for any $\alpha<4$, it fails: there is spontaneous magnetization at low temperatures (see, for instance, this paper).

Concerning your second question, I don't think that there is any way of making the Mermin-Wagner theorem works in systems of dimension genuinely larger than $2$.