"Drama" can also mean being dramatic in real life. Getting into fights, for example. If you are in high school, and you and your best friend get in a long argument, which lasts for several weeks, that can be referred to as drama. What the article means is that Obama is not engaging when people say things about him. He is attempting to remain rational, without responding in ways that may be poorly-thought out or emotional.
For example, he hasn't directly responded to the fact that the Republican party plans to sue him, because he doesn't want to get involved in the drama around that.