No, we are not playing God.
Science is not religion, religion is not science. Medicine is science. Doing things with science does not threaten religion, which is pretty much the basis for people's fear when they say that man is "playing God."
We learned how to harness fire. We invented the wheel. We channeled waters in order to irrigate crops. We rotated crops. We built tools. We built dwellings. We created roads, first as simple paths. We built waterways, and bridges, and dams. We created communications mediums (written, cross-location, etc), we invented machines to propel us, first on land, then in the air. We broke the sound barrier. We broke free of the bonds of this earth. We advanced medicine to the point that we live more than 2 to 3 times longer on average than just 100 years ago. Our bodies have devices implanted in them. We can sustain life, medically, almost indefinitely through the aid of machines.
Are we playing God through this new advancement? No, of course not... no more than we were when we figured out that the Earth revolves around the Sun.
Man learns. Man advances. He changes himself and his environment. That was God's plan...for us to have dominion on this sphere, to master it, to make it ours, to do with it what is our will.