From what I've historically seen, your site won't go down instantly, but you'll owe monies to Magento that they have every right to collect on, and you may not get access to download software updates, etc. Since EE (Magento Commerce) users are typically large businesses, I think that Magento has taken a reasonable stance, but I haven't seen this tested since they were acquired by Adobe.
I also haven't seen this happen often, so I can't give a definitive answer of what happens in every instance. Most folks that are going to renew their licenses do so in advance so that they don't feel pressured, and can negotiate (ex. discounted tickets to Magento Imagine). Those that are migrating away from EE do so in a timely fashion in order to avoid licensing fee headaches. Historically, licensing is annual, so you really don't want to go over into a new year if you plan on switching platforms.
Rest assured that for USD $20k+++, Magento is going to be reaching out and following up about renewal. 
