Yes, Twitter does scan a tweeted URL. It needs to do that to determine if page contains Twitter Card markup. If I remember right trying to detect and block malicious links was also implied when they started to forcefully wrap all links in their own shortener.
See URL Crawling & Caching for details on their bot.
That said the bot hardly does anything that should knock servers over, if it wanted to. As per comment you should be digging through server logs for starters.