No, MC runs fine without Admin rights I believe. However, you have to make sure you have proper access to any media files that you want MC to manage. If you are using local media, this is pretty simple. Select the PARENT folder of the files you want to check, right-click, choose Properties, and check the proper permissions for the user account you want to use MC from.
However, for network usage, ALL file shares are typically read-only for non admin users on Windows, unless you've explicitly granted that user permissions. Windows has three "tiers" of file permissions:
1. The actual file permissions. These are similar to the POSIX U/G/A permissions except that they are a bit more flexible. You set these locally (not over the network) by right clicking on a file or folder, and selecting Properties. To apply the permissions recursively, you need to do it via advanced. Please note that Windows will NOT apply permissions to any folder that you don't "own" (and will silently fail) so you might need to check the Ownership tab under advanced as well.
2. The network share permissions. These apply only to users accessing the file share over the network. They are SEPARATE FROM the file-level permissions, but both apply. So, if the local user has permission to read/write the file share, but read-only local access to the actual file or folder, then it will operate as effectively read-only. These default to read-only when you create a new share.
3. The file attributes. This mainly impacts the file "read-only" attribute. Note that if you view whole folders full of files, the "read only" box will almost always be "partially checked" on Windows Vista/7 because of hidden read-only files in the directory structure. If you uncheck this box manually, it will unset the read-only attribute for any files and folders that you own and have proper write permissions for recursively. But, when you close and re-open the dialog, it will usually show "partially checked" again (because of system-owned hidden files that have read-only status).