Few in I have few dollars left is being used as an adjective to qualify dollars. It is a way of stressing that one's dollar supplies are low.
In a few, few is being used as a noun, hence the presence of the article. Idiomatically I have a few dollars left does not stress the limitations of one's dollars, but indicates more positively that one has some dollars left.
The different circumstances in which each might be used could be as follows:
few dollars left - I cannot possibly stay in America any longer, I have few dollars left.
a few dollars left - Hey, I've got a few dollars left, let's go out to dinner this evening.