I'm not a tax professional, but AFAIK this would have no effect on your current tax situation. The IRS just classifies people as tax residents or not- citizenship or even documented/undocumented status doesn't matter. (The difference is that permanent residents and citizens are automatically tax residents, whereas there's a 183-day residency rule for all other people.) So your U.S. tax burden wouldn't change at all.
Well, what about Australia? Like all sane countries, they don't tax income of people who don't live there. So no issues with that.
In fact, there are only two countries that do tax non-resident income- the U.S. and Eritrea. So the only way this could affect your tax situation is if you move abroad (to Australia or elsewhere) and renounce your U.S. citizenship to avoid continuing to pay those taxes.