The Great Eye In Dol Guldur?

[Legate of Amon Lanc]

Interesting question and first, I have to say that I'd like very much to explore this possibility. But, I have to swiftly (and unfortunately, for this idea of yours, which I like nevertheless) add the facts which make the answer quite clear.

I'll try to sum it up clearly. The main thing is, unfortunately, that the story will not end well. The fate of Middle-earth is quite dark in this case.

Originally Posted by Sardy

How would the trilogy be changed? How would the Fellowship's mission be affected?

My answer is: no trilogy, and no Fellowship (and no mission).

Now I'd better show why. So, here we go:

1. Sauron remains in Dol Guldur. However, by that time he has already sent all the Ringwraith to reoccupy Mordor (around the year 2000. The attack on Dol Guldur happened in 2851). And even if he did not until this time, he certainly would later. He wanted Mordor. He just didn't want to be too obvious. But when prepared, he'd return to his lands. Thus,

Originally Posted by Sardy

One thing that occurs to me right off is that Mt. Doom would be much more accessible, and thus the Ring easier to destroy...

certainly does not come to possibility.

2. Even if Mordor itself were not guarded. We know that Sauron wanted (or at least Gandalf think so, and quite logically) to attack Rivendell and Lórien. I'll include a larger part from the Unfinished tales to make it completely clear:

I thought then, and I am sure now, that he wanted to attack Lórien and Rivendell, as soon as he was strong enough was his original plan. It would have been a much better plan for him, and much worse for us.
You may think that Rivendell was out of his reach, but I did not think so. The state of things in the North was very bad. The Kingdom under the Mountain and the strong Men of Dale were no more. To resist any force that Sauron might send to regain the northern passes in the mountains and the old lands of Angmar there were only the Dwarves of the Iron Hills, and behind them lay a desolation and a Dragon. The Dragon Sauron might use with terrible effect.
Often I said to myself: "I must find some means of dealing with Smaug. But a direct stroke against Dol Guldur is needed still more. We must disturb Sauron's plans. I must make the Council see that.'
(...)
Resistance still had somewhere where it could take counsel free from the Shadow. How could the Ringbearer have escaped, if there had been no Lórien or Rivendell? And those places might have fallen, I think, if Sauron had thrown all his power against them first.

So, here you have it. I think no further explanation is needed. If Sauron wasn't kicked from Dol Guldur, he'd attack the Northerners, then Lórien, Rivendell... and even if Bilbo could have returned back to the Shire (and Bolg's army under the Mountain wouldn't be boosted with a few creatures from Dol Guldur), he'd surely see the Ringwraith in the Shire himself many years sooner than Frodo did. And followed by a horde of Orcs.