How do opponents of penal substitutionary atonement theory handle Isaiah 53? How do opponents of the penal substitution theory of atonement handle Isaiah 53?
This seems to describe penal substitution quite well:

53:4 Surely he took up our pain
and bore our suffering,
yet we considered him punished by God,
stricken by him, and afflicted.
5 But he was pierced for our transgressions,
he was crushed for our iniquities;
the punishment that brought us peace was on him,
and by his wounds we are healed.
6 We all, like sheep, have gone astray,
each of us has turned to our own way;
and the Lord has laid on him
the iniquity of us all. (Isaiah 53:4-6, NIV)
Well I am an opponent of penal substitution and can give an answer that I believe is within the bounds of Eastern Orthodox tradition.

Note the part where the verse says: "yet we considered him punished by God". This phrase implies that it is wrong to consider him punished by God, which is explicitly what penal substitution teaches in the reformed tradition. He voluntarily took our pain; God was not punishing him because he needed some satisfaction. The iniquity was laid upon to accomplish the purpose of transforming man, not of fulfilling some need God had.

Notice that it says "by his wounds we are healed", and not "by his wounds God is satisfied". I personally think this is a very strong argument against substitution theories, because it only talks about Christ's work as healing us, not satisfying God.

My favorite quote to illustrate a more historical perspective of the atonement (as already done by myself here: What is the difference between &quot;ransom theory&quot; and &quot;penal substitution&quot;?) is from Gregory Nazianzus: 


  But if the price is offered to the Father, I ask first of all, how? For it was not the Father who held us captive. Why then should be blood of His only begotten Son please the Father, who would not even receive Isaac when he was offered as a whole burnt offering by Abraham, but replaced the human sacrifice with a ram? Is it not evident that the Father accepts the sacrifice not because he demanded it or because He felt any need for it, but on account of economy: because man must be sanctified by the humanity of God, and God Himself must deliver us by overcoming the tyrant through His own power


I think the most damaging versions of substitutionary atonement are those that claim Christ's righteousness is a substitute for our own. I think a correct way to explain it is that Christ descended and "took our place" so that we might rise and take his. I think a deeper understanding of "taking our place" lies in understanding it as communion and not as replacement. 

To end on a quote that may offend my Protestant brothers and sisters: St. Athanasius of Alexandria once said 


  "The Son of God became man, that we might become god" 


(https://orthodoxwiki.org/Theosis), notice how this view doesn't really fit with a substitutionary view, because we are to actually become like Christ, not just accept his righteousness as a substitution.