In addition to the five choices in which there seems to be a lot of agreement (Canadian Rocky Mountain Parks, Kiliminjaro National Park, Machu Picchu, Sagarmatha National Park, and Swiss Alps Jungfrau-Aletsch), I'd like to place a couple more votes for mountains that also fulfill other categories:

Huascaran National Park -- this site covers the diversity of the Andes with its unique fauna and flora
Nandi Devi and Valley of Flowers -- this site covers unique Himalayan flora, and includes some Himalayan fauna
Western Ghats -- this is a significant geographical component of the Indian subcontinent, and, as kkanekahn mentions, is rich in biodiversity

Wrapping up this one. Have now also counted the half-votes/maybe's and the additional comments, to come to a somewhat broader selection.
Will bring these over to the general list at http://www.worldheritagesite.org/allbyrank.php

Canadian Rocky Mountain Parks - range of aspects: continental divide range, fossils (Burgess), lakes/glaciers etc
Canaima National Park - Highest Waterfall, "unique" formations and biota.
Cliff of Bandiagara
Dolomites
Durmitor National Park
Golden Mountains of Altai
Great Himalayan National Park - W Himalayan flora and fauna
Huanglong - For its travertine features
Huangshan - scenic beauty
Huascaran National Park - diversity of the Andes
Kilimanjaro National Park - highest mountain in Africa, strato volcano, giant flora etc
Machu Picchu -
Maloti Drakensberg
Meteora
Mount Kenya - strato volcano, giant flora etc similar to Kilimanjaro
Mount Sanqingshan
Mount Taishan
Nanda Devi and Valley of Flowers - Himalaya, alpine flora
Pirin National Park
Putorana Plateau
Pyrénées - Mont Perdu - visually dramatic landscape, alpine flora
Rwenzori Mountains
Sagarmatha National Park - highest in world and many other high peaks, glaciers etc
Simien National Park
Swiss Alps Jungfrau-Aletsch - representing the Alps, high alpine scenery, largest glacier in Eurasia
Tajik National Park
Waterton Glacier International Peace Park
Western Caucasus
Western Ghats - rich in biodiversity on the Indian subcontinent
Wulingyuan - sandstone karst landscape
Xinjiang Tianshan

Nanda Devi and Valley of Flowers is still not in the recommended list.

Done!