OR-208 was designated at the inception of the Oregon state route numbering system in 1935 along the Farmington Highway #142, which started at OR-10 in Beaverton and passed through the communities of Hazeldale and Farmington before ending in rural Washington County at OR-219. This route remained pretty much the same until 1982, when an urban renewal in downtown Beaverton took out a railroad and rerouted several city streets. As a result, the Farmington Highway was extended to reach the Beaverton-Hillsdale Highway, creating a continuous road connection between the two. OR-10 was subsequently rerouted from SW Broadway to the Farmington Highway, whose jurisdiction within the city limits of Beaverton was transferred to the city. Some maps, such as Thomas Guide maps, kept misidentifying the road as OR-208 well into the 1980's after OR-10 was extended.
As far as I can tell, there is no indication of OR-208 that remains on any former section of the highway, though any former indication that exists now would be a great roadgeek find.
What was OR-208 between 1935 and the 1982 still exists today as OR-10 from the Farmington area to Beaverton. One tiny portion that is no longer a state route or under state maintenance is a 120 foot-long section of Watson Ave. in Beaverton, shown in red. This short section of former OR-208 provided the highway's connection OR-10. This street is now a one-way street, only allowing someone to travel south from SW Broadway to OR-10.