We know that use of azathioprine and other thiopurines is associated with a small but statistically significant increase in the risk of developing lymphoma, as shown in  and several other easily findable studies.
Do we know by what mechanism(s)?
Some of my speculations that may aid an answer are listed below:
- Thiopurines are cytotoxic. They directly cause damage to cells and or DNA which then leads to lymphoma.
- Thiopurines suppress the immune system, which makes one more susceptible to viruses, the viruses then cause damage to cells, which leads to lymphoma.
- The types of people given thiopurines, such as those with inflammatory bowel disease are at higher risk of developing lymphoma anyway, even without drugs, due to their disease.
- Some other reason not given here.
- Some combination of the above.
An ideal answer would reference studies and highlight which of these factors we have most evidence for, and / or which are the most significant factors.
As a side note, Anti-TNF agents such as Infliximab and adalimumab have also been shown to be associated with higher risk of lymphoma. As far as I know, these drugs are antibodies and are not cytotoxic. Which would seem to imply that 2. is a stronger factor.