Ising model is defined as lattice model with interactions only between nearest sites if lattice.
If we deform Ising model, include non-nearest interactions or interactions between more than two spins, we will stay in same universallity class.. At least naively...
Also, according to Landau, properties nearly critical points are unlikely determined by symmetry and dimension of space.
How one must deform Ising model to obtain another universality class?
I doubt there is a completely exhaustive answer to your question, but one way to change the universality class of the Ising model is to introduce long-range interactions. In particular, consider adding a term $$ H = H_{\mathrm{ising}} - J \sum_{i \neq j} \frac{\sigma_i \sigma_j}{|\mathbf{r}_i - \mathbf{r}_j|^{\alpha}}, $$ where $H_{\mathrm{ising}}$ is the usual nearest-neighbor Ising model, and $\sigma_i = \pm1$ are the Ising variables. Let's just consider the model on a $d$-dimensional hypercubic lattice, and assume that $\alpha>d$ so that the total energy is extensive. It is well known that if $\alpha$ is large enough, the system stays in the usual Ising universality class. However, if $$ \alpha < 2 + d - \eta_{\mathrm{SR}}, $$ then the transition is instead described by a different, "long-range Ising" universality class (actually a family of universality classes whose critical exponents vary continuously with $\alpha$). Here $\eta_{\mathrm{SR}}$ is the anomalous dimension of the operator $\sigma$ in the short-range critical Ising model, $$ \langle\sigma_i \sigma_j \rangle \propto \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|^{d - 2 + \eta_{\mathrm{SR}}}} $$ for $|\mathbf{r}_i - \mathbf{r}_j| \gg 1$. The original references for this are Fisher, Ma, Nickel, PRL 29, 917, (1972), Sak, PRB 8, 281 (1973) and Sak, PRB 15, 4344 (1977).
