Producers don't always burn sets immediately upon cancellation, but I don't know how the determination is made whether to keep a set intact and continue to pay for its storage. Based on what little I know:
Alex Trebek once said that when Battlestars was first cancelled, NBC advised them not to burn the set, hinting that they might bring it back.
The original FF set was stored outside under a tarp on the ABC Prospect lot, where some of the wooden pieces became slightly warped. That same set was used later for the Ray Combs version of the show taped at CBS. The Ferranti-Packard end game board and its associated control equipment were kept. The set was used for the last time in an Old Navy commercial and then destroyed.
When NBC cancelled HS in 1980, Heatter-Quigley took the show to Las Vegas but NBC wanted to sell the set to H-Q for an outlandish sum of money (so the story goes) and it was cheaper for H-Q to build an entirely new set, including the game board. It's surprising that H-Q didn't own the set by then, which presumably had been built for the CBS pilot in 1964.
The Tattletales set was burned after its first cancellation and a new one, slightly modified, had to be built in 1981.
Generic reusables such as scoreboard readouts are kept, but the custom-built controlling electronics would be discarded.