They'd likely be any of the following: insanely moody, difficult or impossible to reason with, extremely defensive & sensitive, have very poor communication & social skills, self-absorbed, cynical, live in a filthy mess of chaos, be obsessed with minor details they've blown out of proportion, passive-aggressive, have an explosive temper (maybe even violent), a major detachment from reality, seem confused & lost in life, be withdrawn, rebellious, depressed, have an inferiority complex, seem arrogant as a defense, and may isolate themselves from people. There might be substance abuse also.
Besides the crazy cat lady, there's also the self-absorbed, reclusive, miserable artist stereotype for unhealthy INFPs (unlike the ISFP who is romanticized for this, the INFP is outcast). Another stereotype might be the incoherent hippy who lives in a pigsty and gives out philosophical advice to teenagers. Then there's the overly passive INFP who puts up with other people's abuse and seethes internally, but likely they explode and become one of the above stereotypes, or wilt into the crazy cat lady.
However, please keep in mind that what may seem like "problems" in an INFP to an ESFP may not be problems at all. Much of the natural INFP behavior and perspective goes against the mainstream flow of society, but that doesn't make it unhealthy.