Is there a way to ask user for Camera access after they have already denied it on iOS?

After some research it looks like you can't do what I'd like. Here is the alternative that I coded to pop a dialog and open the Settings app automatically if on iOS 8+.

Some notes:

• Since iOS 10 you need to specify NSCameraUsageDescription key in your Info.plist to be able ask for camera access, otherwise your app will crash at runtime.
• Once the user changes any permissions for your app, it will kill your app. Handle accordingly and save any needed data before the user hits that "Go" button.
• At some point between iOS 8 and 11, Apple no longer required the user to touch the Privacy cell in the Setting apps in order to get to and change the Camera settings. You may want to change your instructions on what the user is supposed to do in the Settings app based on what iOS version they are using. If someone wants to leave a comment below telling us all what exact iOS version that changed in, that would be awesome.
• As of the last edit of this answer, the code below is working on iOS 14.2.

Swift 5.2:

At the top of your view controller:

import AVFoundation

Before opening the camera view:

@IBAction func goToCamera()
{
    let status = AVCaptureDevice.authorizationStatus(for: .video)
    switch (status)
    {
    case .authorized:
        self.popCamera()

    case .notDetermined:
        AVCaptureDevice.requestAccess(for: .video) { (granted) in
            if (granted)
            {
                self.popCamera()
            }
            else
            {
                self.camDenied()
            }
        }

    case .denied:
        self.camDenied()

    case .restricted:
        let alert = UIAlertController(title: "Restricted",
                                      message: "You've been restricted from using the camera on this device. Without camera access this feature won't work. Please contact the device owner so they can give you access.",
                                      preferredStyle: .alert)

        let okAction = UIAlertAction(title: "OK", style: .default, handler: nil)
        alert.addAction(okAction)
        self.present(alert, animated: true, completion: nil)
    @unknown default:
        fatalError()
    }
}

Denial alert with completion block:

func camDenied()
{
    DispatchQueue.main.async
    {
        var alertText = "It looks like your privacy settings are preventing us from accessing your camera to do barcode scanning. You can fix this by doing the following:\n\n1. Close this app.\n\n2. Open the Settings app.\n\n3. Scroll to the bottom and select this app in the list.\n\n4. Turn the Camera on.\n\n5. Open this app and try again."

        var alertButton = "OK"
        var goAction = UIAlertAction(title: alertButton, style: .default, handler: nil)

        if UIApplication.shared.canOpenURL(URL(string: UIApplication.openSettingsURLString)!)
        {
            alertText = "It looks like your privacy settings are preventing us from accessing your camera to do barcode scanning. You can fix this by doing the following:\n\n1. Touch the Go button below to open the Settings app.\n\n2. Turn the Camera on.\n\n3. Open this app and try again."

            alertButton = "Go"

            goAction = UIAlertAction(title: alertButton, style: .default, handler: {(alert: UIAlertAction!) -> Void in
                UIApplication.shared.open(URL(string: UIApplication.openSettingsURLString)!, options: [:], completionHandler: nil)
            })
        }

        let alert = UIAlertController(title: "Error", message: alertText, preferredStyle: .alert)
        alert.addAction(goAction)
        self.present(alert, animated: true, completion: nil)
    }
}

Objective-C:

At the top of your view controller:

#import <AVFoundation/AVFoundation.h>

Before opening the camera view:

- (IBAction)goToCamera
{
    AVAuthorizationStatus authStatus = [AVCaptureDevice authorizationStatusForMediaType:AVMediaTypeVideo];
    if(authStatus == AVAuthorizationStatusAuthorized)
    {
        [self popCamera];
    }
    else if(authStatus == AVAuthorizationStatusNotDetermined)
    {
        NSLog(@"%@", @"Camera access not determined. Ask for permission.");
        
        [AVCaptureDevice requestAccessForMediaType:AVMediaTypeVideo completionHandler:^(BOOL granted)
        {
            if(granted)
            {
                NSLog(@"Granted access to %@", AVMediaTypeVideo);
                [self popCamera];
            }
            else
            {
                NSLog(@"Not granted access to %@", AVMediaTypeVideo);
                [self camDenied];
            }
        }];
    }
    else if (authStatus == AVAuthorizationStatusRestricted)
    {
        // My own Helper class is used here to pop a dialog in one simple line.
        [Helper popAlertMessageWithTitle:@"Error" alertText:@"You've been restricted from using the camera on this device. Without camera access this feature won't work. Please contact the device owner so they can give you access."];
    }
    else
    {
        [self camDenied];
    }
}

Denial alert:

- (void)camDenied
{
    NSLog(@"%@", @"Denied camera access");
    
    NSString *alertText;
    NSString *alertButton;
    
    BOOL canOpenSettings = (&UIApplicationOpenSettingsURLString != NULL);
    if (canOpenSettings)
    {
        alertText = @"It looks like your privacy settings are preventing us from accessing your camera to do barcode scanning. You can fix this by doing the following:\n\n1. Touch the Go button below to open the Settings app.\n\n2. Turn the Camera on.\n\n3. Open this app and try again.";
        
        alertButton = @"Go";
    }
    else
    {
        alertText = @"It looks like your privacy settings are preventing us from accessing your camera to do barcode scanning. You can fix this by doing the following:\n\n1. Close this app.\n\n2. Open the Settings app.\n\n3. Scroll to the bottom and select this app in the list.\n\n4. Turn the Camera on.\n\n5. Open this app and try again.";
        
        alertButton = @"OK";
    }
    
    UIAlertView *alert = [[UIAlertView alloc]
                          initWithTitle:@"Error"
                          message:alertText
                          delegate:self
                          cancelButtonTitle:alertButton
                          otherButtonTitles:nil];
    alert.tag = 3491832;
    [alert show];
}

Delegate call for the UIAlertView:

- (void)alertView:(UIAlertView *)alertView didDismissWithButtonIndex:(NSInteger)buttonIndex
{
    if (alertView.tag == 3491832)
    {
        BOOL canOpenSettings = (&UIApplicationOpenSettingsURLString != NULL);
        if (canOpenSettings)
            [[UIApplication sharedApplication] openURL:[NSURL URLWithString:UIApplicationOpenSettingsURLString]];
    }
}

Once they have denied camera access, the user can authorize camera use for your app in Settings. By design, you can't override this in your own code.

You can detect this case with the following sample code and then explain to the user how to fix it: iOS 7 UIImagePickerController Camera No Image

NSString *mediaType = AVMediaTypeVideo; // Or AVMediaTypeAudio

AVAuthorizationStatus authStatus = [AVCaptureDevice authorizationStatusForMediaType:mediaType];

// The user has explicitly denied permission for media capture.
else if(authStatus == AVAuthorizationStatusDenied){
    NSLog(@"Denied");
}

For Swift 3.0

This will lead the user to settings for changing the permission.

func checkCameraAuthorise() -> Bool {
    let status = AVCaptureDevice.authorizationStatus(forMediaType: AVMediaTypeVideo)
    if status == .restricted || status == .denied {
        let dialog = ZAlertView(title: "", message: "Please allow access to the camera in the device's Settings -> Privacy -> Camera", isOkButtonLeft: false, okButtonText: "OK", cancelButtonText: "Cancel", okButtonHandler:
            { _ -> Void in UIApplication.shared.openURL(URL(string: UIApplicationOpenSettingsURLString)!)}, cancelButtonHandler: { alertView in alertView.dismissAlertView() })
        dialog.show()
        return false
    }
    return true
}