If there’s a patent on a socially-useful item, or even if not useful, and I decide to spend my vast billions creating something that “violates” the patent (i.e., no license, etc.) but just give it away instead of profit from it (like a consumer product), am I still violating the patent? What about if I were even to prove that the market for items made by the patent are not hurt by my “give away” item (let’s say it’s to people who wouldn’t normally buy it at market price)?
It’s still an infringemnt of the original patent holder’s rights, even if your infringing invention is done without profit motive and/or personal use. 35 USC 271(a) establishes the stautory basics for patent infringement. However, do note subsection (e) which does not allow for infringement where it “is primarily manufactured using recombinant DNA, recombinant RNA, hybridoma technology, or other processes involving site specific genetic manipulation techniques) solely for uses reasonably related to the development and submission of information under a Federal law which regulates the manufacture, use, or sale of drugs or veterinary biological products.”
The damages an infringee can go after, IF he proves you knowingly infringed, are not limited to your (perhaps nonexistant) profits. He can go after the profit he would probably have made, even if you weren’t making that much.
AFAIK if you don’t know you’re infringing (which is NOT the hypothetical case the OP suggests), you couldn’t lose anything. Though, of course, after you’re accused of infringing, you can’t claim ignorance if you continue.
Not that I have plans for doing this, but just wondering. So, if somebody infringes but can show that they make no profit from it (i.e., they give it away and absorb all the other costs like a charity) do they end up paying a fine? I guess just paying a fine also assumes that you can show you did no damage to the holder which might be hard but I assume if you’re making something that the recipients would never be able to afford, you could have an argument; assuming (still more!) that they don’t sell it on the market themselves thus undermining the patent holder’s market. Alternatively, maybe the holder is not using it for anything (yet) and you can say until they do, you haven’t interferred with any profits of theirs and haven’t created any for yourself. 
Oh, I forgot, standard disclaimers apply, you’re not my client, this isn’t legal advice, blah blah blah…
There are still statutory damages found notably in 35 USC 284 Damages, which are expanded a bit more clarified throughout Title 35. These other damages include: a reasonable royalty, interest, and court costs, possibly attorneys fees, and then treble damages.
Patent rights are normally quite different from copyright rights, but both do use the same underlying principle: that of control.
As long as you hold the patent or the copyright, you and you alone have sole legal rights to decide what to do with the item. You can make it available, or, conversely, hold it back until you think the market would be better suited for it.
No matter what you do, it is your decision. When somebody else comes along and says for you, I’m usurping your legal control and imposing my will on your product, they are in a heap of legal trouble. The monetary damages may depend on the amount of money they made or the amount of money they have denied you for making but the underlying legal principle they are violating has nothing to do with the money. It’s breaking the legal control.
Nothing in your scenario alters this. You have done irrevocable damage the minute you created and marketed your product based on someone else’s patent. After that, it’s just an issue for the lawyers exactly what the punishment will be.
Damages available include lost profits, or a reasonable royalty, with treble damages and attorneys’ fees if your infringement was willful. If you’re giving the invention away to people who could never have bought it from the patentee, they may not be able to make a good showing of lost profits, but they can still get you with the reasonable royalty. In addition, they (usually, in light of the recent Ebay decision) can get an injunction forbidding you to do it any more, in which case you’ll be in contempt of court if you keep it up (which could end you up in jail).
Also, contrary to the suggestion above, you can be liable for patent infringement even if you have no notice of the patent. However, generally speaking, you cannot be held liable for willful infringement (treble damages and attorneys’ fees) if you are unaware of the patent. (Recent decisions also suggest that you cannot be held liable for inducement of infringement if you are unaware of the patent, but I’m not sure that’s settled doctrine yet.)