What are the prospects for long distance backpacking trails being inscribed as World Heritage Sites? The Route of Santiago de Compostela is already on the list, but what about other trails such as the Appalachian Trail? If so, what trails have the best potential to be World Heritage Sites?

US Backpacking Trails (Triple Crown)
Pacific Crest Trail (includes Yosemite)
Continental Divide Trail (includes Yellowstone)
Appalachian Trail (includes Great Smokey Mountains)

US National Historic Trail
El Camino Real de Tierra Adentro
Selma to Montgomery National Historic Trail

Korea Baekdu-Daegan Trail (North Korea - South Korea)
Mount Baekdu to Jirisan National Park (includes approximately 6 Tentative Sites)

Chances for inscription? The US backpacking trails would be extremely unlikely. The Camino Real Trail is a realistic proposal to complete the extension of the Mexico nomination with Santa Fe. Indeed, this trail is certainly possible and may be included in the US tentative list update in 2016. In addition, the Selma to Montgomery National Historic Trail may fit into the proposal for a Civil Rights WHS that was discussed for the 2008 US Tentative List, and which might be included in 2016. The Baekdu Daegan Trail has been proposed and may eventually become a tentative nomination as well.

Not sure for the question do you mean nature hiking trails or cultural/pilgrimage routes? If hiking trail, winterkjm did provide a list, but if you mean cultural/pilgrimage route, the next big projects maybe Via Francigena from Canterbury to Rome, or other extension of Silk Road in Uzbekistan or India, Maritime Silk Road from China to Southeast Asia, or Route from Phimai to Angkor in Thailand and Cambodia or another extension of Routes of Santiago de Compostela in Portugal.

Durian

Both, but I was particularly interested in the prospect of nature hiking trails

In terms of the hiking trails you mentioned I think the chances are pretty much nil.

The routes that are already inscribed are there not because they a great hikes rather it is due to their historic role over a thousand or so years, being important factors in the way faith was expressed. And they have a fair amount of built structures along them.

Alas I don't think the Appalachian Trail could stretch to that, it seems like a great hike though.

Winterkjm is right in suggesting the Camino Real may be the best bet, and Civil Rights/ Underground Railroad could have potential if handled correctly.

Having spent a fair bit of time in Utah recently I'm slightly inclined to think there is a pretty viable Mormon migration route WHS lurking somewhere.
There is a story of westward migration and manifest destiny, 19th century religious faith and tales of hardships and persecution (both of and by the LDS). More importantly there are very tangible built aspects to it, not just in Temple Square in Salt Lake City but also the trail has lots of churches, tabernacles and quarters along it. Could be viable but slightly of the original topic.

What is the likelihood that any routes of a great exploration would become a World Heritage Site?

Would today's famed hiking routes (e.g., Appalachian) have any greater value 100 years from now to the World Heritage Committee?

Parkitecture might, which is located throughout most US National Parks, many National Monuments and National Trails.