Sorry about the delay in writing this bit. One of the hazards of having a life is that it sometimes gets in the way of blogging ;)
Christ on the Cross - Traditional Views
The key for any theological answer to the question of whether God suffers is how it handles the crucifixion of Jesus. Of course, if God can't suffer and Jesus is really God and suffers, there's a problem.
Arius used that problem as one of the main reasons he decided Jesus wasn't God, and it was an important challenge for the Church. The answer they came to eventually was that Jesus was one person, both fully human and fully divine, and that his human and divine natures were distinct. So Jesus suffered in his human nature, but not in his divine nature. Christians could even say that God suffered in the humanity of Jesus.
On one level, that's a really important statement to make. I was rightly asked after my last post about pastoral responses to suffering, and one really important point to bear in mind is that God does suffer in the person of Jesus. He isn't a God who is immune from suffering.
Problems with the Traditional Answer
However, the Greek philosophy underlying us talking about Jesus as having two distinct natures has pretty much fallen away. We can rightly say that he is fully human and fully divine, but I don't know anyone who could actually tell me what it means for God to suffer in the person of Jesus but not in himself.
In addition, there are plenty of references in the Bible to God seeming to suffer because his people are suffering or because they are sinful.
How can I give you up, Ephraim?
How can I hand you over, Israel?
How can I treat you like Admah?
How can I make you like Zeboiim?
My heart is changed within me;
all my compassion is aroused.
I will not carry out my fierce anger,
nor will I turn and devastate Ephraim.
For I am God, and not man—
the Holy One among you.
I will not come in wrath.
Hosea 11:8-9, NIV
It seems so much simpler to say that God can and does suffer, though that suffering is because of us and because he chooses to love us. Choosing to love someone often leads to pain, especially when they aren't perfect...
But that then raises the questions about how an eternal God can suffer in time, and whether an eternal God suffering means that suffering wins in the end...