Partial Orders on finite sets to which Zorn's lemma does not apply Zorn's lemma is stated in the text I'm reading as

If $(A, \leq)$ is a poset such that every chain of elements in $A$ has an upper bound in $A$, then $A$ has at least one maximal element.

However, I can't think of any partial ordering (defined as a relation which is reflexive, anti-symmetric, and transitive) on which the condition

If $(A, \leq)$ is a poset such that every chain of elements in $A$ has an upper bound in $A$

would not hold if $A$ is a finite set. (I understand how an infinite set like $[0, 1)$ would fail to satisfy this requirement). My argument is that if $C$ is a chain in the poset $(A, \leq)$ then there will be an end to the chain which will be an element in the chain, and hence every set will satisfy this.
Then can we say that if $A$ is a finite set then any poset $(A, \leq)$ has at least one maximal element by Zorn's lemma? If not, is there a counter-example to help me understand?

I actually wrote , many years ago , an example that may satisfy your curiosity:
https://coldoc.sns.it/UUID/EDB/02M/?lang=en_US
Indeed in this example there is a partial order that cannot satisfy Zorn's Lemma hypotheses, but it is not obvious at first sight.
