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Twitter's crawler will respect robots.txt when scanning URLs. If a page with card markup is blocked, no card will be shown. If an image URL is blocked, no thumbnail or photo will be shown.
Twitter uses the User-Agent of Twitterbot/1.0, which can be used to create an exception in your robots.txt file. For example, here is a robots.txt which disallows crawling for all robots except Twitter's fetcher: