It seems the man (and woman) had not gained anything like immortality yet according to Genesis 3:22. I'm re-quoting from the NJPS since it's more understandable:
And the Lord God said, “Now that the man has become like one of us, knowing good and bad, what if he should stretch out his hand and take also from the tree of life and eat, and live forever!”
Whether or not anything died or suffered up to that point is not known. The text of the curse implies that various hardships were not known to Adam and Eve, but it's possible the text means that these were not present in Eden. It's possible they always existed outside of Eden and the way the curse was implemented was verses 23-24:
So the Lord God banished him from the garden of Eden, to till the soil from which he was taken. He drove the man out, and stationed east of the garden of Eden the cherubim and the fiery ever-turning sword, to guard the way to the tree of life.
The critical question therefore is what is meant by Genesis 1:31:
And God saw all that He had made, and found it very good. And there was evening and there was morning, the sixth day.
Thematically it seems like "very good" in this context means without the suffering and pain implied in the curse of found in chapter 3. But it could be that "very good" means that everything God made was well ordered and in it's proper place. If so, there might be pain and suffering, but they would be serving their proper function.
A Philosophical Aside
There's a philosophical question of whether good and evil are opposite sides of the same concept or whether they are distinct ideas which is raised here. What actually happens when "evil" enters a world? If Evil is an entity opposing Good, then it's entirely possible that pain, suffering and death entered into the world at the moment of the fall. On the other hand, if evil is not an opposing entity, but is merely the absence of good, then pain, suffering and death would always have been potentially part of creation. At the moment of the fall, these unpleasant events would start to have negative effects on creation.
An analogy might be to imagine creation before the fall as a perfectly designed Porsche 911 (or perhaps a Toyota Hilux). In the dualist view, God is the driver and Satan gets into the passenger seat and starts interfering with God: pulling at the steering wheel, moving the rear-view mirror, trying to set the parking break, and so on. But the neoplatonic view is that when the fall occurred, creation was broken. It's as if someone pulled a random hose so that the car keeps running, but not well. In the first model, something new and separate from creation causes problems and in the second, these things were always latent in creation until they were activated by the fall.
If we knew for sure which of these models Genesis had in mind, we could make more educated guesses about whether "unpleasant" things such as death, pain and suffering were possible or actual or neither before the fall. But since we see elements of each view, we can't.
A Christian Perspective
Normally I prefer to answer Tanakh questions from the text itself and not rely on Christian sources. But in this case, the answer is too good to pass up. According to the John's vision of the end of this age, the end point of history will be a state without pain, suffering or death:
Then I saw a new heaven and a new earth, for the first heaven and the first earth had passed away, and the sea was no more. And I saw the holy city, new Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband. And I heard a loud voice from the throne saying, “Behold, the dwelling place of God is with man. He will dwell with them, and they will be his people, and God himself will be with them as their God. He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning, nor crying, nor pain anymore, for the former things have passed away.”—Revelation 21:1-4 ESV)
The suggestion here is that the new heaven and new earth will replace the current heaven and earth, which are full of death and pain. Strongly implied by this vision is that the new existence will fulfill the promise of the original creation. Therefore, God's intention before the fall was that there would be no death, pain or suffering.
Genesis does not say for sure that there was no death or pain before the fall, but it seems to be strongly implied that there was not. Certainly God withheld the fruit of the tree of life (which would allow man to live forever) from Adam because he ate fruit forbidden to him.