I basically want to know if there is some known classes of finite posets where all of their elements are joins of join-irreducibles. It naively feels like it should be all, but seeing as I am struggling to find any references to any such statements, I assume I am missing something elementary.
It would also be interesting to know what is known in the infinite case.
This is a revision of an earlier answer, which contained an error.
It naively feels like it should be all
This IS correct. Let's agree on terminology before getting into details.
Let $\langle P; \lt \rangle$ be a poset. Given $p\in P$ and $J\subseteq P$, say that $p=\bigvee J$ ($p$ is the join of $J$ or $p$ is the least upper bound of $J$) if $p$ is the least element of $P$ that majorizes every element of $J$. Call $p$ a proper join of $J$ if $p=\bigvee J$ and $p\notin J$. Call $p$ join-irreducible if it cannot be expressed as a proper join.
Using these definitions, we have:
Theorem. If $\langle P; \lt\rangle$ is a finite poset, then every element of $P$ is a join of join-irreducible elements.
Reasoning. Assume that the theorem is false, that $\langle P; \lt\rangle$ is a counterexample, and that $p\in P$ is minimal among elements of $P$ that are not joins of join-irreducibles. The element $p$ cannot be join-irreducible itself, otherwise $p=\bigvee J$ for $J=\{p\}$, and this would represent $p$ as a join of join-irreducibles. Therefore we may write $p=\bigvee J$ for some set $J=\{j_1,\ldots,j_m\}$ that does not contain $p$. Necessarily each $j_i\in J$ satisfies (i) $j_i<p$, and by the minimality assumption on $p$ we have (ii) $j_i=\bigvee K_i$ where each $K_i$ is a set of join irreducible elements of $P$.
Let's investigate whether the set $K:=\bigcup K_i$ of join-irreducibles has a join. If $k\in K$, then $k\in K_i$ for some $i$, hence $k\leq j_i < p$. This shows that $p$ is an upper bound for the set $K$. If $q\in P$ is any other upper bound for $K$, then $q$ is an upper bound for $K_i$, hence $j_i\leq q$ for each $i$. Since $q$ is an upper bound for each of the $j_i$'s and $p$ is the least upper bound of the $j_i$'s, we have $p=\bigvee J\leq q$. This shows that each upper bound $q$ of $K$ majorizes $p$, so $p$ is the least upper bound of $K$. Thus $p=\bigvee K$ represents $p$ as a join of join-irreducible elements of $P$. \\\