I am a US citizen work remotely for a US-based multinational corporation. I will be visiting Italy for less than 6 weeks while visiting friends and doing research to start my own company. Do I need a work visa in order to work remotely for my US company (being paid in US currency deposited to my US bank account)? My work is only for US clients and in no way involves Italy or EU (other than the fact that I'm here, of course). I am not completely certain, but I believe that they do not have an office here in Italy.
Legally speaking you do.
Practically speaking, if you don't get a visa and don't tell the immigration officers that that's what you plan to do - there's a very low chance of being caught. That said, you will be breaking the law, and if someone tips about you to the authorities - you may be arrested and deported.
I am not an expert, what follows is based only on my understanding of the Italian Fiscal laws
I don't think you have to pay any tax in Italy if you are not resident in Italy for taxation purpose. For that you need to live and work in Italy for at least 6 months.
On the other hand you must pay taxes for what in Italian is called territorialità, that means locality. I don't think this is your case because as you said, your work doesn't involve Italy at all.
Halfway through that page:
regarding people not resident, taxed at the source of their income, their "fiscal residency" is where they produce the most relevant part of their income
(the original part)
per le persone fisiche non residenti, tassate sulla base del principio della fonte, il domicilio fiscale è nel comune il cui è prodotta la parte prevalente del reddito.
P.S. Apologies for the links in Italian language only
Have you heard of PT? http://en.wikipedia.org/wiki/Perpetual_traveler
Who says you can't work online for your home country while on a 90 days visa free stay in Chile, Brazil, etc?
The whole point concers tax. AS you already pay tax in your home country and do not stay more that 180 days a year in one country, you are considered not resident for tax purposes. Check expat forum and the laws in each country. In general any country where you can stay visa free less than 6 months is good. An you being an American, you are the only citizen in the world, apart from Eritreans, that pays tax based on nationality, so even if you wanted not to pay US tax, you could only do that by renouncing your US citizenship.
Read more about flag theory.
All the best!