Not usually. When a new male or males take over a pride, they will kill the cubs of the previous males in order to bring the females back into heat. It takes a lioness around 2 years to raise cubs to independence, and she will not mate again until this time. Since most males only hold a pride for between 2 and 3 years, they cannot afford to wait all this time while the lionesses raise another male's young - by killing the cubs, they are able to mate with the lionesses themselves, fathering their own cubs.
Lions do fight with neighbouring prides on occasion, but mostly avoid conflict by scent-marking and roaring, which announces their claim to the territory. Fighting will also occur when nomadic males enter a territory and try to take over a pride. However, in neither case are the lions specifically trying to kill each other - they are simply trying to drive off their rivals, and usually one will back down before either is badly hurt. Lions are immensely powerful animals, and fighting to kill could result in even the victor dying of its wounds. Of course, sometimes they do kill each other in fights, or die later from their wounds, but this is not a frequent occurance.
Despite what Jim says, there is no dominance order within a lion pride. There is no heirarchy as there is in, say, a wolf pack. At a kill, males often feed before females simply because they are larger and are able to claim 'the lion's share', but it is also common for the whole pride to pitch in and feed all together, with much snarling and squabbling. Where there is more than one male, one does not dominate the other when it comes to breeding rights and so on - when a lioness comes into heat, one male will often lay claim to her and 'guard' her, and the others will simply back off. Sometimes there is a brief squabble over who gets to mate, but one will quickly back down - maintaing their alliance against invading males is more important. Besides, the males are often related, so even if a male does not mate himself, any cubs born will still carry some of his genes. In other cases, all the males will mate with a lioness in heat.