Do dead monsters really have to leave the game?
There are a few things you could do, but they would have a significant impact on the way the game works.

There are 2 ways to win King of Tokyo. Get 20 points, or be the last monster alive. Any suggestion that makes it no longer a game of elimination would remove the second way of winning.

The trick is that if losing all your health doesn't eliminate you from the game, you need to decide what it does mean. There are a few options of how you could handle it though:


If you have no health, you cannot earn any points or attack. Thus the only thing you can do on your turn is heal. So you are still in the game, but you are slowed down a bunch, making it harder to win.
Similar to above; if you have no health, you just skip your turn, and instead automatically heal some given amount of health. This way you aren't relying on continually re-rolling, hoping for health, to get back in. But you are still slowed down as you miss a normal turn.
When the game ends, consider your health points to be part of your score. Thus someone who has no health left will not have as many points. He could still win, but it would be harder.
When you have 0 health, damage takes away your points instead. In this way, your points act kind of like additional health. Having no health is still really bad, because then you can't win by getting enough points either.