Do lions and tigers mate with their siblings and offspring?

Nature has some safeguards in place to minimize this from happening.

Lions are social animals which live in prides. The structure is typically 2 males, 6 or more females and a number of adolescents and cubs. The male adolescents are kicked out of the pride by the 2 males before reaching sexual maturity which minimizes incest from occurring. Should the pride be taken over by other males then again they would be expelled or killed. The females stay with the pride. Generally the males will only control the pride for 2 to 3 years so it is unlikely any female offspring will reach sexual maturity before their Fathers are ousted.

With tigers it is a bit different. They are solitary. A male tiger will have a territory which will overlap that of several females. He may well be top cat for a number of years but providing there is sufficient space for grown up cubs to establish a territory far away then incest will not be a problem. However, with shrinking space in wildlife reserves and the lack of ‘corridors’ between them incest is starting to become a problem in some places and it is becoming necessay to move tigers from overfull reserves into other reserves to keep the gene flow moving.

I feel bound to add that most incest which occurs is engineered by humans trying to breed genetic freaks like white tigers and white lions.