What do you mean you can't prove an opinion false? OF course you can. If you tell me it is your opinion that steel is made of wood, I can very easily prove you wrong. Of course an opinion can be false.
He means to say that you cant prove entirely subjective statements false.
For example: "Punk Rock music sounds awful." <- Go ahead and prove this opinion false.
You cant, and thats the point. It can neither be proven right, nor wrong. Its just a matter of preference, personal taste.
And for the record, your example "your opinion that steel is made of wood" is just a self-contradictory claim. A statement of fact that has an inherent contradiction. I.e. steel cannot be made out of wood, by definition. If it were, it wouldnt be called steel, but wood.
TL;DR; You can only prove opinions false, if they are presented as statements of fact, not if they are presented as subjective opinions that are separate from facts.
I hope this helps.