Graves provides two sources for this story: Apollodoros’ Bibliotheka 3.5.3, and and Pausanias’ Hellados Periegesis 2.31.2. As you can read for yourself, there is no mention what so ever of Hestia giving up Her throne. In fact, the sources only address the part of Graves' text that follows afterwards, about Dionysos bringing His mother Semele up to Olympos as well.

 

So, did Graves lie? Well, yes and no. Graves is a storyteller; he spun stories based on facts he could find. If he could not find a fact, he made it up to fit the story. Because of this, his books are a great read, but they are not reliable as far as ancient mythology goes. As for The Twelve; there was never a set grouping of Them in ancient Hellas, what mattered was that there was a counsel of twelve, the Dodekatheon, at all. Who resided on the golden thrones was subject to debate, and varied per location.

The most canonical version of the Dodekatheon is represented in the relief above, currently located at the Walters Art Museum. The relief dates back to the 1st century BC to the 1st century AD and depicts the Twelve Olympians carrying their attributes in procession: from left to right, Hestia (scepter), Hermes (winged cap and staff), Aphrodite (veiled), Ares (helmet and spear), Demeter (scepter and wheat sheaf), Hephaestus (staff), Hera (scepter), Poseidon (trident), Athena (owl and helmet), Zeus (thunderbolt and staff), Artemis (bow and quiver), and Apollo (cithara). No mention of Dionysos.
Obviously, Theoi who were held in high regard in a certain city-state would have held the thrones, according to the people who lived in that city-state. This means that it's quite likely there were people in ancient Hellas who firmly believed that Dionysos occupied one of the thrones of the Dodekatheon. Most likely, there were also people who believed Hestia did not occupy one of the thrones. It's entirely possible that some people--perhaps even the same people who believed Dionysos was part of the Dodekatheon, but not Hestia--believed that Hestia gave up Her seat to Dionysos. The problem is that there are no ancient sources to support this, and there was most certainly not a wide-spread myth to this effect that held sway in ancient Hellas.

As a Traditional Hellenist, I place Hestia on Her throne, where she belongs, but this does not mean I do not welcome Her into my home, that I feel She does not occupy the heart(h) of the house, that I do not tend Her flame, or that I do not offer to Her first and last whenever I make a sacrifice; She walks amongst mortals for sure, but that does not mean She does not voice Her opinions to the other Theoi from Her golden throne on Mount Olympos. It also does not mean that I feel Dionysos is in any way a lesser deity; He is a beautiful Theos with a wide domain, who was of vital importance to the ancient Hellenes. Just look at the Anthesteria.

Which position you take is up to you, of course, but I feel it is important to know where your believes find their base. I believed Graves' version of the Dodekatheon for years before discovering it was a modern invention. For me, that was enough to change my views, for others it might not be. The ancient Hellenes did not have a consensus on this, so I see no reason for us to come to one either. Still, the facts matter, so here they are.