There actually is. Self-sacrifice may not be common in the animal kingdom but it does occur.
Ant's will intentionally fight to the death, even when retreat is possible, to defend their colony. Such behavior ensures the overall survival of their hive/nest. This means their genes stick around, and spread the "self-sacrifice" gene. This is, in a way, a positive form of self-destruction, with the goal of ensuring the survival of others. However, there is also evolutionary basis for negative self-destruction. This is particularly noticeable in humans.
Humans do not usually have the same environmental pressures (at least to the same degree) acting on us, that animals do. This means that folks with weaker hearts don't get eaten by the lion, and so they pass on the genetic predisposition to heart disease. Combine this with human social psychology, and you have a situation where not only do they not get eaten, but they are shunned by healthier folks, and are more likely to mate with someone around the same overall genetic fitness, which leads to their offspring being around their same general level of genetic fitness. Combine this with the fact that folks are more likely to be poor when they have rolled poor genes, and poor folks are more likely to have more children, and you end up with a higher number of folks who's brains are extremely impulsive, obsessive (in the wrong ways), or erratic, and higher rates of cancer among other things. This isn't a hard and fast rule, with genetically based mental illnesses for example, being exhibited in someone extremely successful and rich, leading them to suicide.
People like Robin Williams have brains that, due to weird quirks of genetics, managed to
override some pretty serious issues that they are suffering, leading them to be more successful and pass on the genetic trait that may lead to them to self-destruction eventually. This kind of weird situation where harmful genetic traits slip by nature are also noticeable in creatures which over-specialize in surviving particular environments.
For instance, plague-level viruses are actually TOO GOOD at what they do. If they are too specialized, they may wipe out the entire species of host that they have come to rely on, and be unable to adapt quickly enough to infect new host species, leading to their own extinction as well. At the time, it seemed like a good idea...
Our early ancestors are said to have suddenly reached a point in brain development that they quickly developed the tools necessary to kill massive herds of animals. They sometimes killed most, if not all in the herd, and drove the herds into the ground through at least lack of genetic diversity. This behavior resulted from our ancestors brains developing such that they had the means, but not the understanding. So there is more than one way for self-destructive behavior to come about genetically.
Another resource which is more specifically targeting this exact topic:
For further research, look into studies concerning increasing rates of cancer, for which there are a million studies. One such example:
About 5-10% of breast cancers can be linked to gene mutations (abnormal changes) inherited from one’s mother or father. Mutations of the BRCA1 and BRCA2 genes are the most common. Women with a BRCA1 mutation have a 55-65% risk of developing breast cancer before age 70, and often at a younger age that it typically develops. For women with a BRCA2 mutation, this risk is 45%. An increased ovarian cancer risk is also associated with these genetic mutations. In men, BRCA2 mutations are associated with a lifetime breast cancer risk of about 6%; BRCA1 mutations are a less frequent cause of breast cancer in men.
For folks with a genetic predisposition to develop breast cancer, their increased survival rate means an increase of the gene in the population over time.