As sumelic notes in a comment above, you need to use an apostrophe (’) when you are indicating the loss of part of a number or word, as with ’16 for the year 2016 or ’twixt for the word betwixt. The answer that user21820 provides covers this aspect of the question thoroughly, so I want to focus my answer on Microsoft Word's autocorrect behavior, which the OP mentions.
The autocorrect behavior arises from the fact that keyboards generally force the same unshifted key to do triple duty—as an apostrophe (’), as an open single quotation mark (‘), and as a single close quotation mark (’). In actual practice, this triple-dipping isn't a problem as long as your software program treats all three characters as neutral vertical marks ('). But by default, many word-processing programs (including the 800-pound gorilla Word) try to anticipate which of the three ways you intend to use that keyboard key, based on the adjacent keystroke choices you make.
So if you hit the ' key between two letter strokes—say, I and m—Word will infer that you want the ' keystroke to function as an apostrophe and will curl the character accordingly (’), yielding I’m. Likewise, if you hit ' after a letterspace and before a letter, Word will infer that you mean to start a quotation and will introduce an open quotation mark (‘); and if you hit ' after a letter but before a letterspace, it will infer that you want a close quotation mark (’) and will give you that.
But Word isn't smart enough to recognize situations where you want to use ' as an apostrophe after a letterspace (as with ’16 and ’twixt), so if you're using Word's default "smart" quotation mark preference, you have to trick Word into thinking that you're using the ' in a way that Word recognizes as appropriate for introducing a ’ rather than a ‘. Perhaps the simplest way to accomplish this (assuming that you don't have any smart apostrophes handy to copy and paste) is to type two letterspaces, then go back between them and type two apostrophes (which will turn into single open and close quotation marks), then type the 16 or twixt or whatever the word is that you want to precede with an apostrophe, then delete the letter space between the quotation marks and it, and then delete the superfluous open quotation mark:
As ‘’ 16 arrived, I looked ‘’ twixt the bars of my prison cell...
As ‘’16 arrived, I looked ‘’twixt the bars of my prison cell...
As ’16 arrived, I looked ’twixt the bars of my prison cell...
It sounds like a disproportionate amount of effort for such a simple result—and it is. But that's the price you pay for working with software that's smart but not smart enough.
Anyway, to return to your specific example,
Dean's List ’15 - ’16
handles the apostrophes correctly, and
Dean's List ‘15 - ‘16
does not, but you'll see the latter form occasionally (especially on websites) because the writers and editors responsible for the text (1) aren't aware that their software has turned the apostrophes into open quotation marks, or (2) don't recognize that ‘ is the wrong punctuation symbol to use in that situation, or (3) can't figure out how to outwit the software and get the result they want.