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](http://tac.mta.ca/tac/reprints/articles/18/tr18abs.html).

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.