No, those concerns are completely independent of each other. Rust's memory safety protects from accidentally accessing the wrong memory within the same address space, while the MMU protects against accessing (accidentally or intentionally) any memory in other address spaces.
In addition, the address translation done by the MMU has many more applications, like swapping, memory-mapped files, shared memory, copy-on-write after fork, or stack guard pages, none of which can be done by software alone.