Taking the finitary simple Lie algebras $L$, like $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, then these locally simple Lie algebras
are known to have no non-trivial finite-dimensional module at all, see the work of Penkov.
In this sense there are many examples. However, if you mean locally simple, i.e., *integrable*
modules of these Lie algebras, then this means that they are the direct limit of simple $L_i$-modules $M_i$.
So we need to find non-integrable modules.

Given an integrable module $M$, the dual module $M^{\ast}$ is integrable if and only if
for any $i>0$ $Hom_{L_i}(N,M) = 0$ only for finitely many non-isomorphic simple $L_i$-modules N.
This should give non-integrable modules as dual modules, i.e., examples of modules not being a direct limit of finite-dimensional $L_i$-modules.