The operating system frees up the memory used by an application when you close the application, but what happens inside an application is completely application-specific. For example, when you close the project, obviously the DAW doesn't need your data in memory anymore but it doesn't necessarily free the used memory to the OS, it's not uncommon for applications to maintain their own memory pool for faster allocations. And sometimes applications have memory leaks which only become more noticeable on low-spec hardware, and those typically only clear up by closing the entire application.
Try it out. If closing the project helps, then it helps. If getting more memory is not an option, there are usually ways to tune things a bit - make sure nothing unnecessary runs on that computer, that there are no memory-hungry themes and such in use etc. The DAW itself might also have options to limit the amount of caching, undo levels and so on.