If I marry for a man with tax issues, can I receive a green card or will that be a problem for me?

Generally, when a U.S. citizen who marries a foreign national, they may succeed in a marriage-based adjustment of status application process in the Immediate Relative visa category notwithstanding that the U.S. citizen spouse owes back taxes or has ongoing controversies regarding taxes. Of course, there are many additional details that impact immigration-related eligibility, including the possibility that the tax issues reflect financial matters necessitating a "joint sponsor." It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant facts, could advise about immigration eligibilities, options and strategies and could offer legal representation in the often quite complex application process.