To take a meaningful continuum limit, essentially, you need to be in regime where your field is smooth enough that a gradient expansion is possible. This is usually acheived by associating a very high energy cost to field configurations that take different values on nearest neigbours in the lattice.
The continuum limit of $O(n)$ models is worked out in Fradkin's book, Field Theories of Condensed Matter Systems. For the Ising model a direct continuum limit is problematic because the discrete values of the spin make it impossible to directly elevate the Ising spin to a continuum field. Usually, any continuum limits have to defined by some sort of coarse graining and working with the resulting mean magnetization. For the Ising model, this is worked out by Milchev, A., Heermann, D.W. & Binder, K. in J. Stat. Phys.44, 749 (1986)
Hope that helps.
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