The dog's helmet was open in the movie so it must have been able to breathe. It was kept by the Collector in a cell. It's not like the dog is going into space any time soon: so, why does it still have a space suit on? Even at the end of the movie post credit, Cosmo was running around outside of its container without issues so we know the cell did not contain any kind of special air that the dog needs.
On a side note, I see an issue with potty with that space suit.