I know very little about this, I only picked up on its existence from a Flickr photo. It follows the path which exits the wood so its highly likely its to do with the ironstone mines in Eston.

The track continued in a cutting on the other side of the road, heading off towards Dunsdale, although on closer inspection that is to the north of this area.
A tunnel is marked on this map at Waggonways which also suggest a second tunnel closer to Dunsdale

By way of a disclaimer and warning, its located on a scrambler track (possibly on private land?)

It’s likely unsafe, you’ll get very dirty and need a torch.

*Update 2008* Having visited more mining sites subsequently and spoken to other, I now believe this to be actual ironstone workings rather than a rail tunnel. Its unlined and a bord and pillar arrangement of passages is running parallel to the road not just under it, here are some internal shot taken with a better camera.

Making sure to be aware of the tides, the trip between Staithes and Port Mulgrave is a good one, unfortunately I forgot the memory card in the camera, so only phone photos this time.
Old Nab has the remains of ‘bord and pillar’ working now clearly visible on the shore

Along much of the northern part of Brackenberry Wyke exposes the ‘Pecten’ Ironstone seam, with the fossil shells the seam is named after covering the area.

A large adit into ironstone seams is visible in the cliff at the north end of the Wyke, these workings once extended up to 1km inland although now they are collapsed after a very short distance, the cliff above is also very unstable with stones constantly falling.

A smaller collapsed adit is visible just a few meters south of the larger one

Much further to the south, just before the large landslip known as Sheepstones, there is another large adit.

However this is a good distance up the cliff and its impossible to tell if it travels from any distance, it would seem extremely unlikely given the fragile nature of all the cliffs in this area.