History gives little reason to think the Templars survived beyond 1307. Some who escaped the destruction of the order went to Spain where they joined other orders of Christian knights. Some probably ended up in the Knights Hospitaliers. But we have no history showing they continued as an intact organization of Templars.
Their tradition certainly continued, and many claim they migrated to Britain where they joined the forces of Robert DeBrus in his battle against the English. Supporters of this theory point to Roslyn in Scotland as evidence of a Templar presence. Since underground soundings of Roslyn indicate an open chamber under the church, and it has never been explored, there is quite a bit of room for conjecture.
The connection to Scotland leads others to suggest they were instrumental in the founding of Freemasonry. The Masonic figure of Hiram Abif goes back to the construction of Solomon's Temple, and the Templar origins share that same temple. (Just a few thousand years apart.)
There are a number of fraternal and philanthropic organizations today that claim aTemplar heritage, and some claim a direct lineage back to the first Templars.
So, there is quite a bit of speculation, and in The Templar Concordat I have taken every advantage of that in writing a novel where the Templars not only survived the 1307 attack of the King and pope, but grew and prospered all the way down to today. That's the beauty of fiction.