Consider a (additively written) magma $\mathbf{M} = (M,+)$ with a compatible partial order $\leq$, i.e. for every $m, n, p \in M$ such that $m \leq n$, we have $$ m + p \leq n + p,\quad\quad p + m \leq p + n. $$ Suppose that $\mathbf{M}$ is complete w.r.t. arbitrary joins, i.e. for every subset $S \subseteq M$ there is a least upper bound $\sup S \in M$. For every $A, B \subseteq M$ define $A + B := \{a+b\ :\ a\in A, b\in B\}$.
It is always true that $$ \forall A, B \subseteq M.\ \sup(A+B) \leq (\sup A) + (\sup B). $$
Moreover, when $M = [0,\infty]$, we also have $$ \forall A,B \subseteq M.\ \sup(A+B) \geq (\sup A) + (\sup B).\tag{*} $$
However, I see no reason why the latter inequality should hold in general.
Has property $(*)$ been given a name in the literature? Have algebraic structures that manifest this property been given a name (maybe not p.o. magmas, but p.o. semi-modules or p.o. vector spaces)?
According to Theorem 3.10 and Corollary 3.11 in Residuated Lattices: An Algebraic Glimpse at Substructural Logics, pages 148-150, the property you call (*) holds iff the magma is residuated.
In they're multiplicative notation this means $$x \cdot y \leq z \Leftrightarrow y \leq x\backslash z \Leftrightarrow x \leq z/y,$$ for some operations $/$ and $\backslash$ that can be defined.
Also from the same theorem, these are $$x \backslash z = \max\{ y : xy \leq z \}$$ and $$z/y = \max\{ x : xy \leq z \}.$$ So these are just divisions on each side.
With your additive notation, the correspondent are subtractions on each side.
If these can be defined, then the join is compatible with the operation; otherwise, it isn't.
Edit
After a comment from the OP, I realised it isn't exactly as stated.
The correct form of the result above is, from Theorem 3.10, if the magma is residuated, then property (*) holds, while the converse is true, by Corollary 3.11 if additionally the magma is complete.