Well the tax consequence already applies as a green card holder, unless she's willing to give up her GC once you move.
But you, as a USC, will still be taxable with your world income in the US no matter where you live.
If both of you live in the US right now, I suppose your wife holds a gc?
Assuming your wife qualifies for US citizenship (check USCIS website for eligibility info
https://www.uscis.gov/sites/default/files/files/article/chapter4.pdf)
If your wife wants to be a DUAL citizen, she needs to apply for a "Beibehaltungsgenehmigung" first BEFORE doing anything on the US side.
If she just applies for US citizenship without receiving this BBG, she'll loose the German citizenship.
Here's more reading on that on the official German site:
http://www.germany.info/Vertretung/usa/de/05__Dienstleistungen/02__Staatsangehoerigkeit/02__Beibehaltung/__Beibehaltung.htmlThere's also tons of resources online, including Yahoo and FB groups, where people, who have gone through the processes (both BBG and USC) help each other out with tips
and sample documents etc.
Feel free to message me if you you want links, I know several folks who have gone through these recently. I'm sure they'd be happy to give her pointers.
As others have said, the EU is a huge patchwork of countries with very different health systems and other circumstances, so it's hard to give specific advice unless you narrow it down to a couple of countries that you are considering.