state = {
  paused: false
}

componentDidMount() {
  AppState.addEventListener('change', this.onAppStateChange);
}

onAppStateChange = change => {
  if (change === 'background') {
    this.setState({ paused: true });
  }
}

render() {
  return <Video style={...} source={...} paused={this.state.paused} />;
}