"...following the holy fathers, we all with one accord teach men to acknowledge one and the same Son, our Lord Jesus Christ, at once complete in Godhead and complete in manhood, truly God and truly man, consisting also of a reasonable soul and body; of one substance with the Father as regards his Godhead, and at the same time of one substance with us as regards his manhood...recognized in two natures, without confusion, without change, without division, without separation; the distinction of natures being in no way annulled by the union, but rather the characteristics of each nature being preserved and coming together to form one person and subsistence, not as parted or separated into two persons, but one and the same Son and Only-begotten God the Word, Lord Jesus Christ" - Council of Chalcedon (A.D. 451).
"for in no wise is the Godhead, being by nature impassible, capable of suffering." - Theodoret (A.D. 393-457)
and some bible verses for the Evangelicals in the crowd:
"Every generous act of giving, with every perfect gift, is from above, coming down from the Father of lights, with whom there is no variation or shadow due to change." - Epistle of St. James (1:17)
"I the Lord do not change" - Malachi 3:6
I get why people like the idea of God suffering and changing. It makes him human, it gets us out of having to explain why bad things happen, and it goes against Tradition (which is usually reason enough to do something in academia). It's easier to say God suffers with you than it is to say: God doesn't owe you anything and can kill you and your race if he wants (or 'allow' it if we really want to bring in an active passive distinction).
Philosophically if God can suffer he can change. Change implies imperfection, meaning God isn't perfect if he changes. If he isn't perfect he isn't eternal. If he isn't eternal he is contingent and depends on something beyond himself to survive/exist. This means he isn't sovereign. etc, etc.
This is why I believe God doesn't suffer or change. Certainly Christ in his human nature could change and suffer, but not his divine nature. This isn't an exciting belief, it's medieval, Traditional, boring, but it's the only God I can comprehend or understand.
But of course, if you aren't Catholic, Orthodox, or Anglican, you're free to deny the authority of Tradition and the Ecumenical Councils of the Church, in which case you can say anything you like about the divine nature, including that it suffers and that it changes.