The following examples are taken from non-Forgotten Realms settings.
In the Ravnica setting: House Dimir
House Dimir is described in Guildmasters' Guide to Ravnica (p. 45) as follows:
House Dimir is Ravnica’s dark secret: behind a facade of respectable messengers and reporters lurks an association of spies and assassins whose existence is barely suspected by the populace at large. Secrecy is both House Dimir’s best weapon and its best defense, and much of the guild’s work is hidden even from other members. Dimir agents leave no trace of their covert activities, warping the memories of witnesses to their crimes and even wiping their own minds to remove any evidence of their completed assignments.
The first guildmaster of House Dimir was a vampire named Szadek, whose organization agreed to serve as Ravnica’s couriers, information brokers, and librarians. But Szadek also used his guild’s expertise at gathering information to build a vast spy network, and before long, the secret operations of the guild vastly outweighed its overt ones. Eventually the guild disappeared into the shadows, and most Ravnicans came to doubt its existence.
Their goals are described thus:
To House Dimir, knowledge is power. The guild hungers to learn everything it doesn’t already know, especially the weaknesses of its adversaries, and to exploit those weaknesses for its own gain. Conversely, the house holds its own secrets tightly, because it doesn’t want its enemies to turn the tables. The Dimir lurk in the shadows, methodically gathering the knowledge they need to remake Ravnica to their advantage.
This one seems to be spot on thematically with modern conspiracies about the Illuminati.
In the Exandria setting: the Cerberus Assembly
Possible Critical Role Campaign 2 spoilers below.
In Explorer's Guide to Wildemount (p. 41), we have the Cerberus Assembly. They are not exactly a secret society, their existence is well known (and feared), but they are described as being incredibly powerful with near universal influence:
In the centuries since, the Cerberus Assembly has subtly ingrained itself within most facets of the empire’s government, becoming a force of political and military power that many say could rival the king himself. This reality is not lost on the royal council, and tensions often simmer between the assembly and the Crown. The mages of the assembly are known for their extravagant lifestyles and intimidating history of political and economic control, and much of the populace mistrusts them, though no one dares to cross them or the king.
Further, their goals seem aligned with the modern lore of the Illuminati:
The assembly plays the long game, working to subtly increase its power across the empire and beyond. Popular decisions are often publicly presented as the will of the assembly, while conflicted issues are tailored to seem purely the “will of the Crown,” driving a wedge between the two bodies. The assembly seeks to write history to favor its members as the architects of modern law and culture, with the eventual goal of subverting the power of the king, turning him into yet another puppet of the assembly’s goals. (EGtW, pg. 41)
But it gets better! While the Cerberus Assembly may not themselves be a secret society, they do have their own secret operations task force, the Volstrucker (EGtW, p. 202):
The Volstrucker are a clandestine organization of arcane assassins and enforcers in the service of the Cerberus Assembly. Their operatives bear no official title, but are referred to in hushed tones as “scourgers” by residents of large Dwendalian cities such as Zadash and Rexxentrum.
You are a Volstrucker agent. Your duty is to silence dissidents who would undermine the will of King Dwendal—and more importantly, the will of the Assembly.
Their commander's recruiting process is described (EGtW, p. 42):
Trent [Ikithon] only returns every few years to collect young students for his experiments in the mental conditioning that he calls “awakening.” Many of these students go mad and are locked away, but those who endure become zealots for the assembly and join the Volstrucker, an elite group of arcane thugs commonly known as Scourgers, who perform the assembly’s dirtiest work under Trent’s direction.
The Cerberus Assembly bear strong thematic similarity to the Illuminati in terms of their goals and influence, and their Volstrucker or Scourgers get us to conspiracy theory territory.