The Roman Forum is reported to have already been falling apart in 8th century by a anonymous traveler, according to Wikipedia. The question is, with the Visigothic and Byzantine conquests of Italy, when did the forum cease to play any political role or otherwise when did it start to fall into disuse and disrepair? I expect this to be a process rather than a point in time, so please provide as many details as possible.
The Roman Forum was initially constructed in the 8th century BC (as a temple to Vesta), started hosting games sometime around the 4th century BC, and was continually rebuilt and upgraded until about 29 BC. So it can be fairly said that it was (somewhat organically) designed to service the entertainment needs of the capital of the Roman empire, home to somewhere under 250,000 people.
The problem after that is the city itself went into decline (some might say eclipse). By 350ish AD it was down to no more than 125,000 people. It was no longer the sole capital of the empire, and all the Egyptian grain deliveries that were sustaining its numbers had now been re-routed to Constantinople.
By the mid 6th century, it was controlled by the Ostrogoths, and was no longer larger than Constantinople. By the 8th, it was deemed so worthless that the Pope was left to run the city himself, and it had totally fallen off the map. Here's what Colin McEvedy said about it:
...Imperial Rome, had dwindled into a scattering of villages separated by rubble-strewn fields. Where once Augustus had fed a citizen body 300,000 strong, the Pope was hard put to find rations for a hundred.
I think his "100" here refers to the size of the Pope's retinue, not the size of Rome itself. However, the city didn't make his map for that date, which means it had less than 15,000 people. (It wasn't alone. No city in western Christendom was that size. They call it the Dark Ages for a reason.) It certainly was no longer of a size that required a stadium suitable for a city of 250,000. So basically, it fell into disuse because the entire city did.
A note on the numbers:
I used population figures from by Colin McEvedy (and I believe his partner Richard M. Jones). I like them because I know their source, they are well presented, and I can get them for most parts of the world for most of human history. The biggest drawback to them is that most of the research behind them was done prior to 1980.
There do appear to be another set of numbers for Roman population that are consitently in the neighborhood of 4x higher floating around the internet. I managed to track the source back to a paragraph in Luc-Normand Tellier's Urban World History. Wikipedia uses this in its Rome entry, whereupon it is spread throughout the Internet. Try as I might, I can't suss out where Tellier got his numbers though.
However, since the scale of the difference between the two sets of numbers stays the same, none of this really affects the argument I put forth in this answer, which was based on relative numbers, not absolute numbers.
I'm going to embrace your assumption that by "use" of the forum, we're referring to political discussion. (people doubtless "used" the forum for whatever function they found appropriate at the time). Others may challenge this assumption; I have no doubt that the Roman Forum was used for a variety of religious and public functions even when it was no longer connected with the governance of the Republic/Empire.
I suspect that there are two fundamental shifts that reduce the importance of the Roman forum.
The first shift was probably the crisis of the third century. Governance of the Empire declined, and at one point the army auctioned off the office of Emperor. I have no doubt that politics was discussed in the forum and the Senate during this period, but the point of the crisis was that governance was not extended throughout the Empire in any meaningful fashion.
I think the second and more significant shift away from the political use of the forum was when Diocletian decided that Rome needed to be taught that it was no longer relevant to the governance of the Republic.
[There is a contemporary issue of coins suggestive of an imperial adventus (arrival) for the city, but some modern historians state that Diocletian avoided the city, and that he did so on principle, as the city and its Senate were no longer politically relevant to the affairs of the Empire and needed to be taught as much. Diocletian dated his reign from his elevation by the army, not the date of his ratification by the Senate, following the practice established by Carus, who had declared the Senate's ratification a useless formality. If Diocletian ever did enter Rome shortly after his accession, he did not stay long; he is attested back in the Balkans by 2 November 285, on campaign against the Sarmatians.1
I have heard different stories to explain Diocletian's relationship with Rome, but they all seem to agree that Diocletian made a concious decision to break with history and put Rome and the Senate in their place.
You could make a related but more subtle argument that the evolution of governance into the hands of a professional body of career bureaucrats undermined the function/effect of "public" governance".
You could also make the rather simple argument extracted from wikipedia
Eventually much economic and judicial business would transfer away from the Forum Romanum to the larger and more extravagant structures (Trajan's Forum and the Basilica Ulpia) to the north. The reign of Constantine the Great, during which the Empire was divided into its Eastern and Western halves, saw the construction of the last major expansion of the Forum complex—the Basilica of Maxentius (312 AD). This returned the political center to the Forum until the fall of the Western Roman Empire almost two centuries later. Obviously wikipedia disagrees with my argument above - but I think the disagreement arises from how we interpret the forum falling into disuse.
It was probably around the early 300's AD/CE, when the Roman Forum began to fall into disuse. Emperor Constantine had relocated the Roman colonial Administrative Centers to Constantinople in the East-(formerly, Byzantium), as well as to the city of Milan in Northern Italy. By sapping the Forum of its once mighty reputation and sophistication, cities, such as Constantinople, as well as Milan greatly benefited from Rome's increasing demise. (It should be noted that Constantine's Rome was primarily focusing on the earliest construction of a Church encircling the tomb of Saint Peter.......what would become Saint Peter's Basilica 1000 plus years later).
During the 400's AD/CE The Roman Forum became increasingly susceptible to Visigoth invasions from the Germanic lands to its North-(i.e. Alaric), followed by the arrival of the Mongol-Huns from Central Asia.
When the city of Rome and its Forum collapsed in 476 AD/CE, the once prestigious Roman Forum became a remnant of its not-so-distant glorious past.