There appear to be two candidates which can definitely be considered for oldest road tunnel in the world, and a couple of others which are older but may be disputed. Roughly in chronological order, they are as follows:
1. The Euphrates Tunnel (circa. 2180 to 2160 BC, a little less than 1 km long). This has the most dubious claim as it was probably a passageway or pedestrian tunnel (rather than a road) under the Euphrates for private, royal use. Although nothing remains of it, it appears to have been a remarkable technological achievement and is described by Herodotus and Philostratus of Athens, among others (but details vary, and there may even have been more than one tunnel). This article quotes extensively from ancient sources about how opencast tunneling was used in the construction.
2. The Etruscan tunnel at the Furlo pass (circa. 450 BC, length unknown, but probably less than 40 metres). Umberto Marini, in Gola del Furlo (a brochure published by the Provincia di Pesaro e Urbino) credits the Etruscans with the first ever road tunnel:
In 450 BC, to get past a boulder that hand [sic] blocked the pass
(during the floods of the Candigliano River), the Etruscans – aided by
chisels, water, fire and vinegar – perforated the boulder thus giving
rise to the age of road tunnels
It is unclear if this 450 BC tunnel is the same as the Etruscan tunnel near the later Roman tunnel mentioned in several Wikipedia articles: the Wikipedia pages mention either 3rd century or simply Etruscan times. There are some visible remains but whether it was a proper road tunnel or just a pedestrian passage is also unclear. However, even if the Etruscans cannot claim bragging rights to the first road tunnel, they can certainly claim credit for showing the Romans how to do it.
3. Grotta di Cocceio (aka the Cocceius Tunnel, circa. 38 to 36 BC, almost 1km long). This definitely qualifies as a road tunnel and was constructed under the supervision of the architect Lucius Cocceius Auctus who was also responsible for the Cripta Neapolitana (see below) as well as (probably) a tunnel on the island of Ponza.
It connected Lake Avernus with Cumae and was constructed using counter excavation (from both ends) with shafts. This is the longest road tunnel the Romans ever built.
4. Cripta Neapolitana (circa. 37 BC, 705 metres long). Also definitely a road tunnel, it runs from Naples to Pozzuoli and the building technique used was counter excavation. It was 3 to 4 metres wide and 3 to 5 metres high and made it easier for the Romans to move troops between Naples and the Phlegrean Fields.
By Mentnafunangann (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
Impressive though this tunnel is, Seneca seems to have had a rather dim view of it:
Nothing is longer than this dungeon, nothing more gloomy than these
torches, which don’t let us see through the darkness…even if the
location had light, the dust would swallow it up…