This is a similar problem to "as well as him" or "as well as he".
In "She is older than he is", we consider "than" as a conjunction introducing a subordinate clause, of which "he" is the subject.
If we remove "is", we can consider what remains in two ways:
a) as an abbreviated form, with "is" understood, so that "he" remains the correct form.
b) with "than" as a preposition followed only by a pronoun, which is therefore an object: "him".
I personally would say "older than him". "Older than he" doen't sound wrong to my ear, but seems rather literary and archaic.