Please pardon me for using English and there are some medical terms you may need to understand.
Let's go back to the question. It depends on what type of pain-killer you are taking long-term.
In general, we can divide pain-killer into three common types: paracetamol, NSAID and opioid.
For the first type, which is paracetamol, example is Panadol, Acetamol, Paramol, etc. These medicine won't do much harm even if you take it long-term, provided that you are not exceeding the recommended daily dose, which is 8 tablets of 500mg. In case your exceed the dose, your liver cells may be affected and in severe case, like more than 7g a day, it is lethal (ie it kills).
For NSAID type of pain killers, eg Ponstan, Indocid, etc, these drug may have various effect to your stomach, kidney, liver and heart. The most common side effect is gastric bleeding which is manifested as stomachache and black stool. In rare case, it may cause oedema ( due to effect to kidney), raised liver enzyme ( due to effect to liver) and abnormal ECG (due to effect to heart). But I should reassure you that these medicines are very safe and side-effect is rare.
For opioid pain-killer such as morphine, the most common side-effect when taking long term is of course dependence. It means you need more and more medicine to achieve the same level of painless state. Other common side effect include constipation and sleepiness.