It kind of depends on what you mean by "obsolete". Ancient military developments were not of a clear-cut technocratic sort, where an F-16 is obviously better than a Spitfire. There are a lot of different ways I could answer your question.

In one sense, Hellenistic tactics never became obsolete. There is nothing fundamentally wrong with the concept of having a strong infantry battle line to pin the enemy down, and a strong force of shock cavalry to break him. The combined arms forces fielded by the Hellenistic kingdoms represented the culmination of a long development of organised warfare among Greeks, Macedonians and Successors, in which warfare became more professional, soldiers became more disciplined, armies became more adaptable to different tactical situations, and generals became learned experts on military affairs. The Successor kingdoms mastered the tactical system in which their forces operated, and knew how to make the most of the forces at their disposal. No part of this was ever simply rendered obsolete by a better system. The concepts and achievements of Hellenistic warfare were fairly universal, even if the tools of war have changed a lot over time.

In another sense, Hellenistic tactics were obsolete from the moment Macedonian commanders began fighting each other over the remains of Alexander's empire, shortly after his death in 323 BC. If the point of Alexander's methods was to overawe the enemy with the speed, discipline, drill and bravery of his forces, that point was lost when his armies started fighting opponents of the same kind. The Successors knew each other's tactics so well that battles often became symmetrical, with neither side being able to seize an advantage. Victory became a matter of sheer numbers, or of the slight superiority of a particular contingent over its counterpart in the enemy army. In this warfare among evenly matched opponents, the tactics of the Successors could not bring decisive results.

In yet another sense (and I think this is what you're actually getting at), Hellenistic tactics began to lose their effectiveness as an instrument of Hellenistic expansion against and domination of other communities roughly around the beginning of the second century BC. In the 190s BC, the Romans first defeated Philip V of Macedon, and then twice crushed Antiochos III of the Seleukids. Much has been made of how this was due to the superior tactical flexibility of legionary infantry in pitched battle, and there are indeed few prominent examples of Hellenistic forces defeating Romans in open engagements. That said, the Romans were terrified of the Macedonian pike phalanx, as several sources tell us; it would be wrong to claim that the battle of Kynoskephalai (197 BC) made Hellenistic tactics "obsolete" rather than simply "not invincible".