The games are inherently unfair. Sometimes players attempt an action the gamekeeper doesn't allow - such as hiding. The gamekeeper then employs field elements to cause the player to abandon their plan and come back into play. This happens several times during the first game.
Further, the gamekeeper is responsible for keeping the games interesting. Haymitch knew that if either Katniss or Peeta were to survive, they would have to gain sponsers, and to do that they would have to play into the public's hearts. He hatched the plan that they would appear as a couple to increase their odds of surviving during the game. They played the part and obtained needed critical sponsorship at points during the game.
This captivated the public so much that the gamekeeper decided to play on that - and changed the rules to allow a couple to win the game. Ultimately Katniss and Peeta were the final contestants, and then the gamekeeper reversed the ruling, demanding that they then kill each other. The people of the republic, though they hung on every part of the relationship, were probably not too dismayed by this. Lovers forced to kill each other - what an engaging story!
At that moment Katniss gambles that the gamekeeper would choose to allow both to live rather than both die, and she and Peeta are able to convincingly portray the "I would rather die than live without you" routine quickly enough that the gamekeeper couldn't use a field element to take one or the other out.
The gamekeeper had an instant to make a decision - allow both to die and have nothing to show on the next tour and sell around the capital, or allow both to live.
President Snow's reason
President Snow's power is tenuous. He keeps the districts in line largely through fear, and the Capital through misdirection and misinformation.
He is able to handle the Districts with force, but the Capital is his source of power - if the citizens there do not like what's going on, they will find a new leader.
The Capital Citizens are under the impression that the Districts volunteer for the games, and that they generally like their lives. They don't delve too deeply into where the Capital gets its wealth from, though they are aware the Districts are poorer, they don't feel this is a great injustice.
When Katniss and Peeta play the part of lovers, and the gamekeeper generously allows them both to live, this is seen as an act of mercy.
If they really aren't lovers, though, and were just playing a part to stay alive and gain sponsors, it would raise troubling questions as to whether they really were volunteers. Further, it would suggest that he wasn't in control of the game. Giving the lovers a chance to make it out together is a rule intended to enhance the tension of the show. Changing it so they have to kill each other enhances the tension. Having them both die, however, wouldn't have worked, and shows the audience that the gamekeeper is no longer in control - the participants are.
If they don't play the part of star-crossed lovers, the Capital audience will understand that Katniss and Peeta beat Snow. They were the ones, in that instant, who were in power. Snow can't allow people to think that this ending wasn't foreseen and scripted by him and/or the gamekeeper.
Snow can't exactly hide them away - they are critical celebrities now, and must appear regularly at functions. If he can control their appearance and their relationship, though - including using them against each other - then he can stabilize the situation.
As long as things can remain stable until he can legitimately eliminate them - in a special hunger games where the gamekeeper won't be so wishy-washy and will take needed action long before it becomes necessary.