There are a few descriptions of such beacon system in Old English as well as Old Norse medieval literatures, and the beacon network system have recently attracted attention from the historians since 1990s. It is no wonder that Tolkien, himself was a linguist as well as a philologist in OE/ ME literatures was familiar well with its description in them.
While I also wrote a short notice on the beacon in this thread, the answer of /u/BRIStoneman in the similar question thread, Were Beacons Like In LOTR Ever Used To Communicate?, is really worth reading (since (s)he is the real expert in this topic itself).
The following is an excerpt of the Saga of Hákon the Good (d. 961), the king of Norway in the middle of the 10th century who had been raised in the court of King Æthelstan of Wessex (d. 939) that I also find it the closest to the description of LOTR as /u/BRIStoneman did in his answer:
'Along with this it was ordered that whenever a foreign army was in levy, beacons were to be lit on high mountains, so that one could be seen from the other. It is said that news of the levy travelled from the southernmost beacon to the northernmost borough in seven nights (the Saga of Hákon the Good, Chap. 20)'.
Translation is taken from: Snorri Sturluson, Heimskringla: History of the Kings of Norway, trans. Lee M. Hollander, Austin, TX: U of Texas P, 1964, p. 113.
Note that I don't take this description of the saga at face value, i.e. don't believe the historicity of such kingdom-widebeacon warning system itself in the 10th century Norway, but similar description of watching beacon system can be also found for the 12th century Orkney/ Shetland in northernmost periphery of the British Isles where the Norsemen had established their rule during the Viking Age.
Reference:
As Gondor was largely based on the Byzantine Empire I don't think we can discount the Byzantine beacon system as a potential inspiration for Tolkien.
The Byzantine Beacon system was created on the 9th century CE, reportedly by Leo the Mathematician. The Byzantine beacons were installed across Anatolia to warn of Arab attacks coming up through the Taurus mountains and particularly through the Cilician gates. The signal would be carried all the way from south eastern Anatolia, to Constantinople (some 450 miles). Modern experiments estimate that one hour would have been enough to cover the entire distance. The beacons themselves consisted of large bonfires on platforms within fortified positions atop isolated hills. Two such stations have discovered at Loulon and Argos.
The most substantial account of the beacons is given by the chronicle of Pseudo-Symeon:
"Leo the Philosopher, who became bishop of Thessalonica, acting as adviser to the emperor Theophilus, made two sorts of timepieces which worked at the same rate. The one he set up in the fortress in Cilicia near to Tarsus and the other was kept in the Palace. They had marked on them against each hour what was going on in Syria. For instance if an Arab raid had taken place, that was against hour 1, if it was war, against hour 2, if there was a general arson, against hour 3, if something else, against hour 4, and likewise against the remaining hours. So if any of the twelve occurrences which had been marked by pre-arrangement should happen in Syria at the hour where the occurrence came they lit one of the beacons there. As watchers were looking out intently and carefully for what was indicated by them, the beacon was immediately passed on from the fortress called Loulon to the men on the hill of Argaias and straightway to those on Samon and on Aigilos, then further to those on the hill of Mamas. Thence Kyrizon, then Mokilos, thence the hill of S. Auxentios quickly gave the news to the palace stewards appointed for the purpose up on the balcony of the Pharos."
P. Pattenden, "The Byzantine Early Warning System"
In Aeschlyus' Oresteia, the beginning of the first play in the triad shows a watchman on a tower waiting for a signal to determine if the Trojan War was completed and won. Upon receiving it, he tells Agamemnon's wife Clytemnestra so she has time to finalize her plans to murder him. It's believed in some circles that the signal the watchman was waiting for was a line of signal fires, similar to what is seen in the LotR movie.
This play was written in the 5th century BCE, and the Trojan War supposedly took place sometime before 1200 BCE. So if that interpretation is correct, then it's a very very old method of communication.
EDIT: I did a quick search for a copy of the lines from the play itself and here are the opening lines 11-15:
"And now, as ever, am I set to mark
When shall stream up the glow of signal-flame,
The bale-fire bright, and tell its Trojan tale-
Troy town is ta'en: such issue holds in hope
She in whose woman's breast beats heart of man."
My ancient Greek is rudimentary at best, but this is a pretty faithful translation.
EDIT 2: I'd like to clarify the lines above are not the only lines referencing a beacon fire. Just the first few.