It's hard to answer without having knowledge on how actually you have implemented your code, I can only give basic idea on this.

You need 3 files for this, as

1. main.html:- where the scoreboard is included.
2. scoreboard.php:- where the scoreboard is shown.
3. updateSB.php:- where the scoreboard is updated.

main.html

<div class="somediv">
  <div class="scoreboard" id="scoreboard">
  <!--This is the division I will load scoreboard dynamically-->
  </div>
</div>
<script>
 //Loading initial scoreboard here
 $('#scoreboard').load("scoreboard.php");
 //Now do ajax calls for '+' '-' buttons (onclick) here
 $.post("updateSB.php",{type : "plus/minus"},
 function(data, status){
   if(status=="success"){
     //Loading scoreboard for every update
     $('#scoreboard').load("scoreboard.php"); 
   }
 });
</script>

scoreboard.php

<div class="scoreboard">
  <div class="header">Some team <?echo $team1;?> vs team <?echo $team2?></div>
  <div class="body">
    some sort of table score here?
    Score :- <?echo $score;?>
  </div>
  <div class="footer"></div>
</div>

updateSB.php

<?
//Here is where you update ur scoreboard
if(isset($_POST['type']){
  if($_POST['type']=="plus"){
  //Perform plus on score
  }
  elseif($_POST['type']=="minus"){
  //Perform minus on score
  }
}
?>

Points to remember

1. Use ajax in proper way ie, do right request and handle the request at backend correctly.
2. Implementation of dynamic scoreboard is possible if you are using some global variables or database at the backend.

For any queries comment down.