What happens when a gateway on the Ripple network defaults or simply vanishes? Do people that got their money from it lose that money, do other gateways honour the IOUs issued by that gateway still, or does something else happen?
Any of those things can happen. Likely, there will be some "hot potato" going on as people rush to offload any IOUs they hold to people who haven't yet gotten the news. It's unlikely other gateways would honor the IOUs unless they were contractually obligated to. This would likely harm their customers as they'd have to make up the losses somehow -- likely with higher fees.
Gateways are in the business of being trusted. So they have a huge incentive to find ways to induce people to trust them. We don't yet know what will happen, but some possibilities include:
Gateways could cross-insure each other. (But this might make you less likely to trust a gateway for fear that it will collapse due to the collapse of another gateway.)
Gateways could be insured by governments or private insurance companies. (They hold other people's fiat currency, so the legal framework for doing this exists in many countries.)
Gateways could post bonds or liens on physical assets that could be seized or sold if the gateway defaults and used to cover their obligations.
Note that this is only a significant issue for people who wish to use the Ripple system to hold fiat currencies as a store of value. It's not a major issue for a merchant who just wants to get paid in dollars as they can use a gateway that electronically deposits the currency in their regular bank account on a daily basis. If that gateway goes out of business, they'd only lose a day or two's revenue and could switch gateways on the Ripple system with no disruption. (You might think losing two days revenue is a big deal, but remember, businesses already pay 2% to take credit cards. That's like losing seven days revenue every year.)
This question actually gets at the fundamental nature of Ripple. Ripple is essentially an online P2P implementation of free banking. Free banking is somewhat similar to how banking worked in the United States for much of the 19th Century (with a few major caveats) prior to the Federal Reserve and FDIC systems, and indeed back then banks would sometimes just vanish overnight. Because of this, banks went to great lengths to emphasize their trustworthiness; for example in my neighborhood in Brooklyn the Williamsburgh Savings bank built a colossal structure (exterior, interior), basically as a way of signaling they were in it for the long haul.
Interestingly, bills of exchange (that is, IOUs) were also broadly traded at the time, and of course are a major feature of Ripple (more explanation here). In a fundamental way, free banking (and Ripple) seek to build a banking system on trust among parties without relying on promises from a centralized authority such as a government or central bank.
Free banking is a popular idea among Austrian economists because it would be a way of removing government intervention from the currency and banking sectors. You can read more about it on Wikipedia.
The same that happens when an exchange or bitcoin business that accepts deposits disappears: users will likely lose their deposits.
Unless the gateway somehow binds their IOUs issuance to a legal contract through some digital signature scheme. In that case, users may be able to sue them or get something from the bankruptcy process of the company.