Firstly R. Aviner's response conflates the question of whether Olmert deserved what happened to him with whether the disease itself is Divine Punishment. Of course no one can know whether an event is actual divine punishment. There is always room to speculate but ultimately we cannot know. However that is besides the point, because the question is not Did G-d Punish Olmert With Cancer but Does Olmert Deserve Cancer.
Parts A. and C. of R. Aviner's response presumes that the question is Did G-d Punish Olmert With Cancer or May We Say That G-d Punished Olmert With Cancer. Both answer a question that was not asked and are there besides the point. That leaves only B. which references hona'at devarim, harming with words.
Much of these discussions go back to Sharon's illness, back then I took a stand against praying for Sharon. They also go back to the death of Yitzchak Rabin. They revolve around the question of the status of someone like Olmert. Is he to be considered a fellow Jew or human being whose repentance we should hope for or one of the community of the wicked hated by G-d who spills the blood of his brethren, one of the evil shepherds who enslaves and sells his people. R. Aviner's answer presumes the former, I presume the latter.
Did G-d give Olmert cancer? That's a question that is impossible to answer definitively. Does a man who is responsible for the death of other Jews and who embodies corruption and deceit deserve cancer? I imagine he deserves something far more than an easily treatable medical problem that will hardly interfere with his plans to give away Yerushalayim Ihr Hakodesh.
Can we seriously talk about harming the feelings of such a man? The Torah we follow is not the Torah of Gandhi. We can exchange quotes and counterquotes all day, but a true Torah view does not set the wicked man who sells his brother into slavery as someone who deserves a hug and a pat on the back in the hopes that he will repent out of love. R. Aviner somehow believes the idea of the generation repenting out of love runs counter to confronting and condemning the wicked in even the harshest possible ways.
When the wicked are punished the name of G-d is upheld in the world because men see that there is justice and the world is not without a justice. It is forbidden to pronounce the name of the wicked without a curse (Mishlei 10:7) and that is why HaRasha or Yemach Shemo Ve'Zichro was generally appended to the names of everyone from Bilaam to Hitler.
יִגְמָר נָא רַע, רְשָׁעִים--ּ וּתְכוֹנֵן צַדִּיק May the full measure of evil be piled on the wicked, King David prayed. This was not an abstract statement, it was a specific prayer against his enemies.
Why indeed do we celebrate when bad things happen to the wicked? Is it merely personal vendettas, a petty desire to see them suffer? David HaMelech answers that question for us in Tehilim 10: 4-12.
The wicked, in the pride of his countenance says 'He will not require'; all his thoughts are: 'There is no God. His ways prosper at all times; Thy judgments are far above out of his sight; as for all his adversaries, he puffeth at them.
He says in his heart: 'I shall not be moved, I shall never suffer adversity.' His mouth is full of cursing and deceit and oppression; under his tongue is mischief and iniquity. He sits in the lurking-places of the villages; in secret places doth he slay the innocent; his eyes are on the watch for the helpless.
He hath said in his heart: 'God hath forgotten; He hides His face; He will never see.' Arise, O LORD; O God, lift up Thy hand; forget not the humble.
We of course can never know if an evil thing that befalls an evil man is a Divine Punishment, but at the same time it is dangerous too be too enthusiastic in proclaiming that it is not and that we should not speak of it. For when we do, we too risk enabling the belief of the wicked that G-d has forgotten and that he will never see. This only emboldens the wicked who might otherwise at least restrain themselves when they encounter a setback, yet when they hear Rabbis talk down the idea that they are experiencing a divine punishment, they only grow more arrogant. "He hides his face," they think, "he will never see. Even the Rabbis admit it."
When we too loudly disclaim any idea that the sufferings of the wicked is the Hand of G-d, we risk proclaiming ourselves that the world has no judge. No we are not prophets, yet a thousand Jewish communities across the world have held Purims long after the of the Prophets that quite often commemorate the death of evil men. Was Stalin's Purim supported by prophecy? No but we can as a general rule assume that the death of the wicked that brings salvation is a salvation from G-d.
At this point Olmert's cancer is not in that category nor do we have any proof that G-d is punishing Olmert for his crimes. Neither do we have any proof that he is not. The question is whether we give G-d the benefit of the doubt or Olmert. Do we spare Olmert's feelings or do we uphold the belief that G-d judges the world and that if the wicked suffer, it is probably by his hand. Which is more important?