What is the correct generalization of this fact? In general, the class of structures of a first order theory should form a category (right?) and it seems that adding or removing axioms should induce left and right adjoints respectively. Are there any constraints that need to be placed on the axioms being added or removed?
It's always true that models of a first-order theory form a category, and that removing axioms gives forgetful functors between such categories of models. They do not have adjoints in general; an important example is the theory and category of fields, which is obtained from the theory and category of commutative rings by adding the non-equational axiom
$$\forall x \in F : x \neq 0 \Rightarrow \exists y : xy = 1$$
(edit: and the axiom $1 \neq 0$.) This induces a forgetful functor $\text{Fld} \to \text{CRing}$ which has neither a left nor a right adjoint. The left adjoint doesn't exist because rings simply do not admit a universal map into a field; for example already the two maps $\mathbb{Z} \to \mathbb{Z}/2$ and $\mathbb{Z} \to \mathbb{Q}$ are incompatible. The "obvious strategy" to try to construct the left adjoint would be to invert every nonzero element, but this often produces the zero ring; even when it doesn't we don't get every map into a field this way, because some maps into fields have nontrivial kernels.
(Funnily enough there's a way to fix this by replacing fields with what are called meadows, in which the inverse axiom is weakened. Another way to do it is to replace $\text{CRing}$ with the category of integral domains and injections; then the left adjoint exists and is given by the field of fractions.)
Everything is fine if the axioms are equational, such as commutativity. So if $T_1 \subset T_2$ are equational theories (Lawvere theories) we get a forgetful functor $\text{Mod}(T_2) \to \text{Mod}(T_1)$ which always has a left adjoint, given by imposing on a $T_1$-model every new equational axiom in $T_2$, and there can be infinitely many of these.
We can also add operations instead of just removing axioms, e.g. $T_1$ could be abelian groups and $T_2$ could be rings. In this case the left adjoint also always exists and the construction also involves freely creating new values of the new operation, e.g. in the above example we start from an abelian group $A$ and produce the symmetric algebra $S(A)$ which is the result of freely creating new multiplications. There could be infinitely many new operations too.
