The answer is no, you only need to buy an iPhone app once to install it on all the compatible devices that you have.
This is interesting because on desktop and laptop computers, you generally do need to buy one copy of a program for each computer (though there are exceptions – Adobe Creative Suite, Microsoft Word, etc. – that include multiple licenses). Some programs even ensure this with copy-protection technology.
No such copy-protection or licensing terms seem to exist for iPhone apps – yet.