Given any class C of similar algebras, Jónsson's lemma (due to Bjarni Jónsson) states that if the variety HSP(C) generated by C is congruence-distributive, its subdirect irreducibles are in HSPU(C), that is, they are quotients of subalgebras of ultraproducts of members of C. (If C is a finite set of finite algebras, the ultraproduct operation is redundant.)
A necessary and sufficient condition for a Heyting algebra to be subdirectly irreducible is for there to be a greatest element strictly below 1.
Hence every finite chain of two or more elements as a Heyting algebra is subdirectly irreducible.
For example, the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras.
There also exists a single finite algebra generating a (non-congruence-distributive) variety with arbitrarily large subdirect irreducibles.