The short answer is that there are numerous types of caching. Some of these are acceptable for a membership site, and some of these are not. This is not a MemberMouse-specific issue, it affects all membership platforms, and most sites that allow a visitor to log in. Likewise, there are numerous types of hosting. Some of these can be used with a membership site, and some are incompatible with them.
There are also a number of caching technologies that save the complete output of the programs running on your server, and just display this output when the pages are accessed. These are called "full page caching" technologies. They can be used on certain kinds of websites. For example, let's say I run a blog. No matter who is reading my blog, they all see the same thing. I can use full page caching, and it's great for performance, because my server is essentially doing nothing, just receiving the requests and replaying the output over and over again.
However, the very purpose of a membership site is to display different things to different people. Since two visitors will see different content depending on what they purchased, whether they are logged in, or a number of other factors, you can't simply save the output from one visitor and use it with another. The program that controls who sees what has to be run, or people will not see what they're supposed to see, and actions that need to happen (like purchasing or cancelling) may not happen the way they're supposed to, because parts of the program that control those actions are being skipped.
So, what do we do about all this? The answer is that we will need to experiment a little bit. Every hosting provider is different. We've created a guide to Configuring Caching on Your Server in order to optimize performance for MemberMouse and minimize issues. At the bottom of this guide is a detailed sample message that you can send to your hosting provider in order to make sure that caching is configured as you'd need.