No. It is well-known that (infinitary) algebraic theories are precisely the theories whose category of models are monadic over $\mathbf{Set}$, and vice versa. See, for instance, Linton's papers in the Seminar on triples and cohomology.

For an example of a non-algebraic theory, consider the theory of fields. This is not monadic over $\mathbf{Set}$ because e.g. the category of fields does not have products.

It is well-known that the theories whose category of models are monadic over $\mathbf{Set}$ are necessarily algebraic (possibly infinitary), and conversely every (possibly infinitary) algebraic theory that admits freely-generated models on every set has a category of models that is monadic over $\mathbf{Set}$. See, for instance, Linton's papers in the Seminar on triples and cohomology.

For an example of a non-algebraic theory, consider the theory of fields. This is not monadic over $\mathbf{Set}$ because e.g. the category of fields does not have products.