Can "stand" be stative?

Those are two extremely different uses of the verb stand.

The first is intransitive with an adjectival predicate complement describing the subject just as you might find with remain or be. It is a figurative use, not literal standing up straight.

The second is transitive with a direct object of eating liver.

These two senses are found far from each other in the OED. The first is the intransitive one:

I. Of persons and animals, in literal and figurative senses. intransitive

9. a. To remain firm or steady in an upright position, to support oneself erect on one's feet. Often in negative contexts. Also with adjective or adv., as fast, firm, stiff.
b. fig. To remain steadfast, firm, secure, or the like. Also with adjective or adv.

And then much later we find the other one, which is transitive:

V. Transitive senses.

• Originating from the conversion of an indirect into a direct object, from the omission of a preposition, or from intransitive uses with cognate object.

59. a. To put up with, tolerate; (to be able or willing) to endure.
    b. Familiarly in more trivial sense (with negative expressed or implied): To reconcile oneself to, be favourably disposed to, feel any liking for (a repugnant or distasteful object).

Your second example sentence could have been equivalently written:

• I can’t stand to eat liver.

• I can’t stand liver.

• I cannot abide liver.