Perhaps the most frustrating thing that could happen when scaling a Fourteener is being forced to abandon the peak. This is, of course, sometimes necessary due to weather or other unsafe conditions, but it should never be the result of one's canine companion. I have compiled a list of the 54 official Fourteeners and their relative accessibility to athletic canines. This is based largely on the class scheme for the main routes up each Fourteener used in Gerry Roach's guide to Colorado's Fourteeners.
Easily Accessible
Mount Antero
Mount Belford
Mount Columbia
Culebra Peak
Mount Elbert
Handies Peak
Mount Harvard
Humboldt Peak
La Plata Peak
Mount Massive
Missouri Mountain
Mount Oxford
Pikes Peak
Mount Princeton
Quandary Peak
Redcloud Peak
San Luis Peak
Mount Shavano
Sunshine Peak
Tabeguache Peak
Uncompahgre Peak
Mount Yale
Moderately Accessible
Blanca Peak
Castle Peak
Challenger Point
Ellingwood Point
Mount of the Holy Cross
Mount Lindsey
Mount Sneffels
Windom Peak
Inaccessible/Not Advised
Capitol Peak
Crestone Needle
Crestone Peak
El Diente Peak
Mount Eolus
Kit Carson Peak
Little Bear Park
Longs Peak*
Maroon Peak
North Maroon Peak
Pyramid Peak
Snowmass Mountain
Sunlight Peak
Wetterhorn Peak
Mount Wilson
*Dogs are prohibited in Rocky Mountain National Park
As a disclaimer, I suggest that you use your own best judgement when hiking with your dog. You know him/her best, and should respond to the signals provided to you.