I’m always fascinated whenever i read about their relationship, but it feels like at some point their relationship must have deteriorated or just lost the intimacy that it once had, because Robert had other affairs and eventually remarried. And he also greatly angered the queen when he accepted the post of governor-general of the united provinces. How close were they by the time of Robert’s death?
To quote from a previous answer of mine,
People largely theorize about Elizabeth not being a virgin in relation to her feelings for Robert Dudley, a close friend she romantically loved and seriously considered marrying after the death of his wife - a death that he was rumored to have caused in order to marry the queen; after he married someone else, she reacted with what was almost definitely jealousy. If she had been going to risk her reputation and possibly even her throne, it would have been for him.
Ultimately, we can't know, as I said before. However, the absence of evidence points toward her not doing that: she would have been surrounded by servants, pages, guards, and ladies of the bedchamber at all times (as your professor said), and if she had done something sexual in front of them or dismissed them all in order to meet with Dudley in private, it's unlikely that they all would have taken it to their graves. That there was never a scandal or an allegation of sexual misbehavior despite the many people who would have had to work to conceal it implies that it probably didn't happen.
Elizabeth was angry with Dudley when he accepted the position of Governor-General of the Northern Provinces (in the Netherlands) early in 1586, because she had personally turned it down herself in order to avoid turning it into an English vassal state and provoking war with Spain, and because she felt that he was setting himself up as her equal. However, we have to be careful when evaluating her fury, because a lot of history books, particularly those aimed at general audience, tend to punch up emotions. She was certainly upset with his decision to take the post, but did she really have the emotional outbursts "furious" (a word frequently used to describe her feelings over this) implies? Did she hold a personal resentment against him for the entire two years of his tenure, until December 1587? I don't believe we know, and it's unlikely.
When he returned to England, he was broke, having put most of his own money into the war he was leading the Dutch against the Spanish in. And he was still very close to Elizabeth, an adviser and confidante - in 1588, she appointed him to an important position in the army leadership to fight against Spain, and he invited and accompanied her to rally the troops at Tilbury for her famous speech. He dined with her very frequently afterward until he left for the spa at Buxton in Derbyshire to take the waters, as he was in failing health, and he died while slowly traveling back to his residence at Kenilworth. Her grief was fairly extreme and she secluded herself for several days, supposedly only coming out when her councillors had the door to her room broken down. She wrote "his last letter" on the last note he wrote to her on his way to Kenilworth and kept it by her bed. It's safe to say that they were still fairly close by that time.