I'd come here with the intention to say there was no intimacy at all in any of Tolkien's writings, but upon further research, I am torn. And somehow it's about the Lady Galadriel again...
While reading this Quora Q&A answer by Thomas Snerdley (a Tolkienite I regard very highly) my mind on the matter was completely changed. While I'd originally agreed with the answer by Ernst W. Adams (another Tolkienite I regard very highly) and his sentiment that Elvish marriages were unshakeable (something I still stand by) and that Gandalf and Galadriel were merely friends and comrades in arms. However, upon reading Thomas' answer I had a bit of a change of heart, which I'll try to lay out here. And while this may be an unpopular opinion I'll try to provide sources where Thomas hadn't.
I believe there was some form of intimate relationship between the two. It was certainly platonic due to Tolkien's religious beliefs and the fact that adultery was a sin and would not happen, but whether that would've been different had Tolkien had different views is anyone's guess.
The two (Galadriel and Gandalf) certainly shared a close bond. As Darth Satan (User 8719) says they likely spent a large amount of time together as Olorin frequently visited the Gardens of Lorien and there spent time with the Elves who'd returned from the Halls of Mandos, and Galadriel had spent time there before the March of the Noldor. It's possible that Olorin had encountered Galadriel, especially given her fame as one of the most beautiful elf-maids, given:
"the light of the Two Trees, Laurelin and Telperion, had been snared in [Galadriel's] tresses.
Given that Olorin "loved the Elves, he walked among them unseen, or in form as one of them, and they did not know whence came the fair visions or the promptings of wisdom that he put into their hearts." It's more than likely they had interacted.
Even without interactions in Valinor, the two clearly have a close connection in Middle-earth. The two "shared the joys and burdens of bearing two of the Elvenrings". They would most certainly have been aware of the other bearer, whether this had an affect on their relationship isn't certain.
She strongly supported Gandalf's case to be the head of the White Council, putting him against Saruman, the eventual leader.
Galadriel indeed had wished that Mithrandir should be the head of the Council, and Saruman begrudged them that, for his pride and desire of mastery was grown great; but Mithrandir refused the office, since he would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons.
The Silmarillion - Of the Rings of Power and the Third Age
Galadriel and Gandalf clearly had a deep mental connection as well. Liberties of such are taken in the Hobbit films, but even in the books we see Galadriel seemingly reading people's mind, and she was able to "sense" Gandalf's death against the Balrog and sent Gwaihir to his aid, and clothed him in white with a new staff.
“That indeed is the command of the Lady Galadriel who sent me to look for you,” he answered... Healing I found, and I was clothed in white.
The Two Towers
The loss of Gandalf seems to take a significant toll on the Lady Galadriel, saying she could no longer see him nor the paths upon which he walks.
'Now tell us where he is; for I much desired to speak with him again. But I cannot see him from afar, unless he comes within the fences of Lothlórien: a grey mist is about him, and the ways of his feet and of his mind are hidden from me.'
The Fellowship of the Ring - Book II, Chapter 7: The Mirror of Galadriel
While reading through The Mirror of Galadriel for this answer I'd noticed something else, Galadriel is quick to defend Mithrandir when her husband, Celeborn, calls him foolish.
And if it were possible, one would say that at the last Gandalf fell from wisdom into folly, going needlessly into the net of Moria.
'He would be rash indeed that said that thing,' said Galadriel gravely. 'Needless were none of the deeds of Gandalf in life. Those that followed him knew not his mind and cannot report his full purpose. But however it may be with the guide, the followers are blameless. Do not repent of your welcome to the Dwarf. If our folk had been exiled long and far from Lothlórien, who of the Galadhrim, even Celeborn the Wise, would pass nigh and would not wish to look upon their ancient home, though it had become an abode of dragons?'
Furthermore, they fought off the darkness that was the necromancer for many a long age together, fighting at the fortress of the Necromancer in Dol Guldur.
Gandalf had at last prevailed upon Saruman and the White Council to attack Dol Guldur
The Return of the King - Appednix A: Annals of the Kings and Rulers
Finally, they left the East together and sailed to the West with Elrond, Frodo and Bilbo, Galadriel having left Celeborn in Lothlorien
Then Elrond and Galadriel rode on... Then Círdan led them to the Havens, and there was a white ship lying, and upon the quay beside a great grey horse stood a figure robed all in white awaiting them. As he turned and came towards them Frodo saw that Gandalf...
The Return of the King - Book VI, Chapter 9: The Grey Havens
Given those points, one may wonder what about the infidelity that seems to come with the face she seems to have a platonic love for Gandalf. Well we know from before the First Age that Míriel died giving birth to Fëanor, and Finwë took a second wife, Indis. While Indis or Finwë lived, Míriel couldn't return from the Halls of Mandos, as an Elf could not have two wives. Finwë, however, pleading with the Valar to allow his Míriel to return to life in return for him joining Mandos' side in his halls states that one may love two.
“It is unlawful to have two wives, but one may love two women, each differently, and without diminishing one love by another. Love of Indis did not drive out love of Miriel; so now pity for Miriel doth not lessen my heart's care for Indis.
History of Middle-earth - Volume X: Morgoth's Ring
In conclusion, I believe that she may have had a platonic love for Gandalf, and although she was married to Celeborn was able to love both him and Gandalf, both differently, but without diminishing the other.
As for Saruman, there is no particular reason before he forces his way to the head of the White Council that she should have disdain for Saruman, and I believe it's likely that she had faith in him as a worthy Maiar, until Gandalf had discovered his treachery.