To address your Q specifically, the organs that are highly susceptible to ADRs are the kidneys, brain, heart and liver.
KIDNEYS: a) there are some very intricate mechanisms for excretion/reabsorption processes in the nephron that are easily affected by toxic substances. These include several enzymes and cell structures that can be hit badly by chemicals.
b) The concentrations of some drugs reached in the kidneys, simply out of the need for excretion, can be several hundred-fold those in the systemic circulation, obviously making this organ susceptible.
BRAIN: The neurons in the CNS are very finely-tuned delicate structures that can be tipped towards failure by external chemicals.
HEART: The mechanisms that control the heart rate and force of contraction are very precise and depend on accurate homeostasis e.g. if the K+ level wanders above the upper limit of most hospital labs of 5 mmoles/L OR if it drops below 3.5 mmoles/L, drastic brady- or tachy-arrhythmias, respectively, can occur. Another e.g. is the prolongation of the QTc interval [on the ECG] that can be caused by some drugs, which can lead to torsades de pointes [multifocal VT], which can be fatal.
LIVER: Because everything absorbed in the ileum first reaches the liver via the hepatic portal vein for first-pass metabolism, this places the liver in a precarious position, in spite of its efficiency in dealing with toxins in terms of oxidation and glucuronidation, etc.
I feel that this brief summary might help to resolve your query..