How do I programmatically slide the UITableView down to reveal the underlying UIRefreshControl

You'll have to manually change the contentOffset of your UITableView yourself. Be sure to account for the contentInset.top. It should be something as simple as:

CGPoint newOffset = CGPointMake(0, -[myTableView contentInset].top);
[myTableView setContentOffset:newOffset animated:YES];

This will do the trick

- (void)beginRefreshingTableView {

    [self.refreshControl beginRefreshing];

    // check if contentOffset is zero
    if (fabsf(self.tableView.contentOffset.y) < FLT_EPSILON) {

        [UIView animateWithDuration:0.25 delay:0 options:UIViewAnimationOptionBeginFromCurrentState animations:^(void){

            self.tableView.contentOffset = CGPointMake(0, -self.refreshControl.frame.size.height);

        } completion:^(BOOL finished){

        }];

    }
}

For Swift 3, this is what I have based on Peter Lapisu's answer:

override func viewDidLoad() {
    super.viewDidLoad()
    self.refreshControl?.addTarget(self, action: #selector(refresh), forControlEvents: UIControlEvents.ValueChanged)
    // ...
}

func refresh(sender:AnyObject) {
    self.refreshControl?.beginRefreshing()

    if let yOffsetTable = self.tableView?.contentOffset.y {
        if yOffsetTable < CGFloat(Float.ulpOfOne) {
            UIView.animate(withDuration: 0.25, delay: 0, options: UIViewAnimationOptions.beginFromCurrentState, animations: {
                if let refreshControlHeight = self.refreshControl?.frame.height {
                    self.tableView?.contentOffset = CGPoint(x: 0, y: -refreshControlHeight)
                }
            }, completion: nil)
        }
    }
}