The heat energy being moved does not contribute to any global warming. The A/C is simply moving energy from inside to outside; the heat just turns around and flows back into the room, and the total stays the same. It's like someone trying to keep a leaky boat from sinking; you dump some water over the side, and it (or some just like it) leaks back in. A certain capacity pump can keep the water level in the boat low enough for the boat to survive. The A/C does the same for our houses.
But...
The energy used to drive the A/C is a different matter. Depending on the source of this electrical energy, there could be a minor contribution to global warming.
If the electricity came from fossil fuels, then the energy is stored solar energy from whenever the fossil fuel was first formed from atmospheric carbon dioxide by plants using sunlight. You could say that the energy absorbed by plants through photosynthesis kept the earth somewhat cooler than it otherwise would have been, and now we're "correcting" this by burning the fossil fuel back to carbon dioxide.
Here in Ontario, much of our electricity historically came from hydro-electric plants, like Niagara Falls. Does this contribute to global warming? Not really... Sunlight (energy) falls on the ocean. Some of this energy warms the water. The rest evaporates some water, lifts it kilometers into the air, and lets it fall as rain on elevated land. As this water flows back to the ocean, it converts its potential energy into kinetic energy, and then into heat. The water flowing away from Niagara Falls is slightly warmer than the water at the top. But, if the water goes through a generator on its way down, electrical energy is extracted, and the water is cooler than it otherwise would be. So, the heat produced in Toronto by using electricity is exactly balanced by the slightly cooler Niagara River. Eventually all the water makes it back to the ocean, and all the original solar energy is converted to heat...
As commented by @AlanSE, the negative effect of the carbon dioxide in the atmosphere greatly outweighs these energy transfer effects...