If you rm the library prior to installing the new one, I think your system will keep the inode allocated, the file open, and your program running. (And when your program finally exits, then the mostly-hidden-but-still-there file resources are released.)
Update: Ok, post-clarification. The dynamic linker actually completely "solves" this problem by passing the MAP_COPY flag, if available, to mmap(2). However, MAP_COPY does not exist on Linux and is not a planned future feature. Second best is MAP_DENYWRITE, which I believe the loader does use, and which is in the Linux API, and which Linux used to do. It errors-out writes while a region is mapped. It should still allow an rm and replace.  The problem here is that anyone with read-access to a file can map it and block writes, which opens a local DoS hole. (Consider /etc/utmp. There is a proposal to use the execute permission bit to fix this.)
You aren't going to like this, but there is a trivial kernel patch that will restore MAP_DENYWRITE functionality. Linux still has the feature, it just clears the bit in the case of mmap(2). You have to patch it in code that is duplicated per-architecture, for ia32 I believe the file is arch/x86/ia32/sys_ia32.c.
asmlinkage long sys32_mmap2(unsigned long addr, unsigned long len,
                            unsigned long prot, unsigned long flags,
                            unsigned long fd, unsigned long pgoff)
{
        struct mm_struct *mm = current->mm;
        unsigned long error;
        struct file *file = NULL;
        flags &= ~(MAP_EXECUTABLE | MAP_DENYWRITE); // fix this line to not clear MAP_DENYWRITE
This should be OK as long as you don't have any malicious local users with credentials. It's not a remote DoS, just a local one.