Turns out Microsoft Edge, unlike Safari, can still open local html files. Discovered in this answer. To wit:

1. Install Microsoft Edge from the App Store
2. Open Files (or whatever file browsing app you like).
3. Open the file, then tap the Share button to send to another app.
4. Scroll across to "More...", choose Edge, and voila!

I do it using the 'Notes' app on my iphone. Quick and free download.

I emailed the file to myself, then opened it on my phone via email. Inside the email, I clicked the html file. This opened it. I then clicked the download button on the top right of my particular iphone. After that I clicked "Add to Notes".

Inside the notes app, my html file is there and also all functions work as there is also javascript involved.

Unfortunately you can't open File-URLs in iOS' Safari-App via [[UIApplication sharedApplication] openURL:] due to sandbox restrictments.

You have 2 options:

a) Upload that file to a server and then open it in the Safari-App with [[UIApplication sharedApplication] openURL:@"http://..."];

b) Embed a Safari-WebView in your app which displays the html-file, like so:

UIWebView *webView = [[UIWebView alloc] initWithFrame:...];
[self.view addSubView:webView];
NSURL *fileURL = [NSURL fileURLWithPath:...];
NSURLRequest *request = [NSURLRequest requestWithURL:fileURL];
[webView loadRequest:request];

self is a UIViewController here.

EDIT

To add a Share-Button with proper logic, you can put the ViewController, that contains the UIWebView, inside a UINavigationController and add a UIBarButtonItem of type UIBarButtonSystemItemAction to it's NavigationBar. If the BarButtonItem gets tapped, you show a a UIActivityViewController, like so:

- (void)viewDidLoad {
    [super viewDidLoad];

    UIBarButtonItem *shareButton = [[UIBarButtonItem alloc] initWithBarButtonSystemItem:UIBarButtonSystemItemAction target:self action:@selector(shareButtonTapped:)];
    self.navigationItem.rightBarButtonItem = shareButton;
}

- (void)shareButtonTapped:(id)sender {
    UIActivityViewController *activityViewController = [[UIActivityViewController alloc] initWithActivityItems:... applicationActivities:nil];
    [self.navigationController presentViewController:activityViewController animated:YES completion:nil];
}

For more Infos on using UIActivityViewController, see NSHipster