The GR131 is part of a long distance trail series spanning all seven of the Canary Islands and is part of the pan-European E7 walking route. The route is well signposted and can be tackled in either direction with the traditional route running East to West. While the route doesn't take you up Teide, it won't fail to dazzle as you climb to approx. 2500 metres and take in the spectacular national park. Always pack plenty of food/water but this route will take you through/close to La Esperanza, Aguamansa, El Portillo de Teide, Vilaflor, Ifonche and Arona.
See below for two fantastic guides to conditions, amenities and kit advice.
Current (unverified) FKT is thought to be 13:01:31 by the author of the Tenerife Trail blog featured in the description. Photos of the elapsed time & distance along with his/her knowledge of the route can be found in the blog linked above.