Why would immortal feel pain ? Why wouldn't they ?
While I don't really agree with DaaaahWhoosh answer, he pointed out that pain is really usefull.
That said, pain is very useful for getting people to stay alive. It
gets you to stop hurting yourself or being hurt, and thus usually gets
you away from further danger.
I think (I may be wrong) that pain is a powerful stimulus that indicates the body that some threshold have been reached. In that way it is just a feed-back from the body that tells, if the situation remains as it is, there will be a risk of damage to the body. (Damaging a body is different from killing it).
Also, pain is really useful for learning processes. For example, in computer science, when you try to teach a neural network something, you have to recompense him when it does good, and punish him when it does not good.
So, I can't find any reason why immortals won't feel pain as it is a useful tool to learn the limits of one's body.
As of emotional pain, obviously I think it necessary for them. I see that trait as part of empathy and what would be immortals without empathy but psychopaths ?
Again, emotional pain is useful to avoid hurting oneself and to understand why it is wrong to hurt someone else.
Two statements bother me from DaaaahWhoosh's answer :
Now, the fact that humans have become immortal means they no longer
have to flee danger
Being immortal just means you can not die. But there still exist conditions they want to flee (like being imprisoned for decades), so danger still exist for them.
All that's left is the desire to have sex, and make/raise babies
I think the opposite. Some unicellular bacteria are "immortal" but can not reproduce. In fact, would it be sustainable over long-term if immortals could reproduce? There would be virtually an infinite amount of children who cannot die and would have children of there own. Considering infinite life time, when would they stop being a child and becoming adult ?