You get what you test for. - author unknown
When you move to the cloud, you leave much of the burden of on-premise applications behind. But one thing you won't leave behind is testing. It's a basic premise of configuration management: when things change, you test.
In terms of testing, nothing changes when you move to the cloud. Every version upgrade, every patch, every whatever-they-call-the-change-this-month, you test. Functional unit test, integration test, system test, performance test, and so on...it's all still with us.
It's all still with us because, in most cases, patches and updates are built for a large number of customers. And there may be variations across that set of customers. Some of those variations will cause changes in results from one customer to another.
At the end of the day, each customer has the ultimate responsibility of making sure their cloud system works for their unique situation. The same system with the same configuration will not work for all customers, any more than the Chevy Spark my wife drives for urban commuting will work for someone whose primary need is an off-roading vehicle. So if Chevy were to make a suspension system change to all vehicles across the board, it may act differently for the off-roader than it does for my wife. So...we test.
It's your system, regardless of whether it's in the cloud or not. And you get what you test for.