Currently, there is, unfortunately, no support for background tasks of any kind. The feature you are referring to would be a background timer. Such a timer is this product pain (a feature request) for react native, you may upvote it to show an increased demand for this feature.
EDIT 12/2016: There is still no real option. You have the Headless JS API since RN 0.33, but it is only for android. Also your App will crash if this runs in the foreground so you have to be careful about using it. Thanks to @Feng for pointing that out.
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