It's Not 2015 Anymore!

JavaScript ES6, 84 76 75 bytes

s=>(q=s.replace(/2015/g,"<s>$&</s>*"))+(s<q?`
*2016. I'm a year early!`:"")

This is pretty well golfed for JavaScript imo.

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This works by taking the input and replacing all instances of 2015 with <s>$&</s>* where $& is the value matched, but in this case, that's always 2015. Then, if the input is different from q, it'll add the *2016... string, otherwise, It'll add a blank string (nothing).

Ruby, 69 bytes

$><<gets.gsub(/2015/,'<s>\&</s>*')
puts"*2016. I'm a year early!"if$&

Jolf, 60 bytes

Two parts, again.

oHρi"2015"+pd"s'$&'*"
oH                    set H
  ρi"2015"             to input string, replaced all 2015 with
           pd"s'$&'    $& crossed out. (pd is an apply tag function)
          +        *" append an asterisk

    v-- unprintable
+H|&?=Hi+S"*2016. I\'m a year early!'"
+H|&?=Hi                            '" add ↓ to H if H != i; otherwise add empty string ('")
        +S"*2016. I\'m a year early!'  a newline plus that string
                                       implicit printing

Try it here! Bonus points for actually crossing out the text and conforming to the output examples? :3