Allow user to select text from UILabel to copy

It is not possible with UILabel.

You should use UITextView for that. Just disable editing using textFieldShouldBeginEditing delegate method.

A poor man's version of copy and paste, if you cannot, or don't need to use a text view, would be to add a gesture recognizer to the label and then just copy the entire text to the pasteboard. It's not possible to do just a portion unless you use a UITextView

Make sure you let the user know it's been copied and that you support both a single tap gesture as well as a long press, as it will pick up users trying to highlight a portion of text. Here is a bit of sample code to get you started:

Register the gesture recognizers on your label when you create it:

UITapGestureRecognizer *tap = [[UITapGestureRecognizer alloc] initWithTarget:self action:@selector(textTapped:)];
UILongPressGestureRecognizer *longPress = [[UILongPressGestureRecognizer alloc] initWithTarget:self action:@selector(textPressed:)];
                [myLabel addGestureRecognizer:tap];
                [myLabel addGestureRecognizer:longPress];
                [myLabel setUserInteractionEnabled:YES];

Next up handle the gestures:

- (void) textPressed:(UILongPressGestureRecognizer *) gestureRecognizer {
    if (gestureRecognizer.state == UIGestureRecognizerStateRecognized &&
        [gestureRecognizer.view isKindOfClass:[UILabel class]]) {
        UILabel *someLabel = (UILabel *)gestureRecognizer.view;
        UIPasteboard *pasteboard = [UIPasteboard generalPasteboard];
        [pasteboard setString:someLabel.text];
        ...
        //let the user know you copied the text to the pasteboard and they can no paste it somewhere else
        ...
    }
}

- (void) textTapped:(UITapGestureRecognizer *) gestureRecognizer {
    if (gestureRecognizer.state == UIGestureRecognizerStateRecognized &&
        [gestureRecognizer.view isKindOfClass:[UILabel class]]) {
            UILabel *someLabel = (UILabel *)gestureRecognizer.view;
            UIPasteboard *pasteboard = [UIPasteboard generalPasteboard];
            [pasteboard setString:someLabel.text];
            ...
            //let the user know you copied the text to the pasteboard and they can no paste it somewhere else
            ...
    }
}

You use create a UITextView and make its .editable to NO. Then you have a text view which (1) the user cannot edit (2) have no border and (3) the user can select text from it.