Scientists discovered having either of the two oils each day can trigger damage to the liver and also made people susceptible to non-alcoholic steatohepatitis (NASH).
NASH, nicknamed 'the silent killer' can lead to scarring of the liver, which can result in cancer.
Sunflower oil is often used while cooking and fish oil is most often consumed through supplements.
The results were concluded by monitoring the DNA of livers of 72 rats fed virgin olive, sunflower or fish oil from birth until they were two.
Virgin oil did not affect the rats livers.
The University of Granada scientists also conducted tests investigating the 'internal stress' levels of animals and their overall health.
Published in the Journal of Nutritional Biochemistry, the findings dismiss previous evidence that suggested sunflower oils could actually treat NASH.
They were both shown to make the rats 'susceptible to NASH', according to study author Professor José Luis Quiles Morales.
'The alterations caused by the long-term consumption of sunflower and fish oils make the liver susceptible to NASH,' he claimed.
Professor Morales warned that it is a 'very serious disease that may act as a catalyst for other liver diseases such as cirrhosis and liver cancer'.
'Virgin olive oil is the healthiest option, which has already been proven in relation to diverse aspects of health,' Professor Morales continued.