What I mean is that presumably a topic such as logic would have, at this point, been so advanced that the ancestral works are unnecessary.
I would read it for pleasure, but have they been ultimately rendered obsolete?
Logic means something different now than what it once did. In classical philosophy one primary branch of philosophy investigated the cosmos, one investigated human life, and a third investigated the forms of reasoning used in the other two: physikē, ethikē, logikē (See Diog. Laert. for variants on this schema.)
When an older writer (or a modern classicist) refers to Aristotle's "Logic" they mean the entire Organon, the first five books of the Aristotelian corpus. These are, in effect, Aristotle's philosophy itself!
Now, since Frege (roughly) there has been an increasing trend to use "logic" to refer to something like predicate logic or propositional logic: in Aristotle this would correspond to his theory of the syllogism (which is summarized here and there but most extensive in the Analytics). Three thoughts:
Yes, in a sense the syllogism is something like a primitive version of set theory so most people interested in math and philosophy aren't going to find Aristotle's syllogism-theory a huge revelation.
However, it isn't the "inner content" of syllogistic inferences Aristotle focuses on - the actual focus is on how syllogisms relate to things that are not syllogisms, including the subjects and predicates they take as arguments, fields of knowledge where syllogistic deductions are valid, licit non-deductive arguments, and illicit deductive fallacies.
All of this machinery is necessary to understand any of the larger issues of Aristotle's physics and ethics - including what subjects are "sciences" (i.e., use deductive inference) - and also the narrower problem of how to interpret difficult passages.
So the problem isn't really that if you don't understand syllogistic logic, you won't understand how to make a deductive inference; the problem is that if you don't understand how Aristotle's theory of the syllogism fits in with Aristotle's theory of the proposition, Aristotle's ontology, etc. (the "Logic") most of the rest of Aristotle will fly over your head. I've done it both ways and Aristotle definitely makes more sense after you've read the Organon.
(And the same goes for many later philosophers who liberally borrow Aristotle's terminology and assumptions even when they profess to be attacking him. See the essay "Aristotle as cuttlefish" for one of the funnier illustrations; historians of philosophical grammar [Itkonen, Padley] are particularly good at documenting this conceptual debt. So the same point applies even to say, Kant; a lot of the terminology and preoccupations will make no sense if you aren't familiar with Aristotle. In fact Kant is a particularly good example; "apodictic certainty" is difficult to define but easily understood once you understand that apodexein = "to demonstrate" [by syllogism], in Aristotle's sense.)
No, Aristotle's logic has not been rendered obsolete or disproved; "modern works still reference/use his logic frequently" (courtesy: V2Blast).
See, for example:
Łukasiewicz, Jan. 1957. Aristotle's syllogistic from the standpoint of modern formal logic. Oxford: Clarendon Press.
Thomas Greenwood. “The Unity of Logic,” Thomist: A Speculative Quarterly Review 8 (January 1, 1945): 457–470.
from this answer to the question "Can all mathematical reasoning be translated into traditional logic?"
I would suggest that a study of Aristotle's dialectic logic is as vital as ever since it models how we think. Most serious mistakes in metaphysics are a consequence of abusing Aristotle's rules and it happens a lot. I would go so far as to say that it is a failure to study the dialectic that causes most logical problems in metaphysics.
The consequences of failing to grasp the rules of the dialectic can be seen all over the place. It causes intractable antinomies and leads to desperate ideas like logical positivism and dialethism.
So I'd say a study of the dialectic and Aristotle's rules would be crucial to any real grasp of philosophy. An ill-considered view of dialectic contradictions leads to the idea that metaphysical questions are not just undecidable but intractable, a view held by most scholastic philosophers.
I can speak from experience. I spent five years thinking that Buddhist metaphysics was a violation of classical logic and was only able to correct this view after reading a book I now regularly recommend, which is Aristotle's De Interpretatione: Contradiction and Dialectic by CWA Whittaker.
But there is a lot of Aristotle that is not important so while his dialectic method deserves careful study this need not amount to much work.