Thank you for all your feedback. I tried everything you all suggested and unfortunately none of it seemed to fix the issue. After further testing I discovered that it wasn't specifically mobile that was the issue, but safari in general.

I have successfully got it to work. The issue was happening due to incorrectly handling the range provided by the browser.

I changed that part of the code from this:

while (!feof($handle) && ftell($handle) <= $range_end) {
    echo fread($handle, $chunk_size);
    flush();
}

To this:

echo fread($handle, $range_end - $range_start + 1);