No, Columbia should not use Ecuadorian territory to fight the FARC, because that would be invading Ecuador's land without just cause. Perhaps the two countries would sit down to talk about reaching an agreement that would allow Columbia to use the land. But it is unfair to go marching in there without permission, since it is Columbia's mess to begin with.
No, Columbia should not use Ecuadorian territory to fight the FARC, because doing so would stir up unrest in the region. The FARC is a Columbian problem. Therefore, it is unfair to violate Ecuador's territory, without Ecuadorian agreement, in order to fight the FARC. Columbia should either seek Ecuador's permission or find ways within its own borders to respond to the FARC.