This question might seem more negative than it is...
I'm looking to travel the world and build my startup on-the-go. Would I need to pay income tax in any country if I don't stay in a country for more than 1-2 months?
My income would mostly come from affiliate networks here in the United States.
Perhaps these are simple questions that your tax agent/accountant can answer? - that way you have only the best intentions possible
Your citizenship determines this condition. For example, if you are a US Citizen, no matter where you live, you ought to report your earnings and depending on the treaty / agreement with the country in which you earn, there might be DTA (Double taxation avoidance) treaty signed with the respective country and unless you are in complete tax heavens, like the cayman islands or so, there is a very remote chance that you can legally not pay a tax.