When we change the license on our codes from CC BY-SA 3.0 to the MIT license, will we be able to copy and reference code from old posts in new posts, or will we only be allowed to provide links to old posts?
No (except your own, unabridged code, or if you have permission from the owner)
Old code will be under CC-BY-SAv3. One of the conditions of using CC-BY-SA is that if you remix, transform, or build upon the material, you must distribute your contributions under the same license1 as the original. But SE will require you to license your code under the MIT license. You can't license your code to SE under the MIT license, if you must distribute your contributions under CC-BY-SAv3. Therefore, no old code will be permitted to be listed, as such will be copyright violation. The only exceptions is if you own the code, or you ask the owner (all of the owners, actually (both the original poster and all the editors) permission.
Just another reason we shouldn't move to the MIT license.
- There are some exceptions, but the MIT license isn't one of them.
My answer to another question equally applies here. Fair use allows you to use copyrighted work in a larger work, if certain conditions are met. You are required to mark the work so that recipients know that it is available under a different license (including no license). It is up to the person making the post to ensure that they are complying with the law.