LiveTime Object Literal
From a once over:


dst stands for daylight saving time, you should comment that
offset I assume is the offset to the UTC (Greenwich time), you should comment that as well
I am ambiguous about declaring rawOffset, dstOffset, timeoutId. I can see the value of self documenting code, but is it worth it ? Not sure. The code works fine if you delete those 3 statements.
Consider falsy evaluations

if (typeof dst !== "undefined") {
  LiveTime.dstOffset = dst * 1000;
}


compared to

if (dst) {
  LiveTime.dstOffset = dst * 1000;
}


or even (a bit hacky)

LiveTime.dstOffset = dst ? dst * 1000 : 0;

Do not declare variables twice like here:

var now = new Date();
var now = new Date(

You can get the timestring this way:

now.toTimeString().substring(0,8);


This basically keeps only the first 8 (hh:mm:ss) characters of the time string
You can get the UTC date value this way:

new Date((new Date()).toUTCString().slice(0,-4) )        


This gets the UTC date, strips off the timezone part and re-creates the date
document.getElementById("current-time"); &lt;- This should be a parameter
As @Pinoniq mentioned, ideally you should write your class/code so that you can run multiple instances.
LiveTime is not a constructor (yet), so it should be called liveTime


This would be my counter proposal:



function liveTime(elementId, offsetSeconds, dayLightTimeSavingSeconds) {
    "use strict";

    var timeZoneOffset = 0,
        daylightTimeSavingOffset = 0,
        element = document.getElementById(elementId),
        timeoutId;
  
    if( offsetSeconds ){
      setTimeOffset( offsetSeconds, dayLightTimeSavingSeconds );
    }

    function setTimeOffset(offsetSeconds, dayLightTimeSavingSeconds) {
        //Convert to milliseconds
        timeZoneOffset = offsetSeconds * 1000;
        daylightTimeSavingOffset = (dayLightTimeSavingSeconds || 0) * 1000;
    }

    function start() {
        if( timeoutId ){
          return; //We already started
        }
        (function increment() {
            //Set the time to UTC time
            var now = new Date((new Date()).toUTCString().slice(0, -4));
            //Add the configured offsets
            now.setTime(now.valueOf() + timeZoneOffset + daylightTimeSavingOffset);
            //Update the HTML
            element.innerHTML = now.toTimeString().substring(0, 8);
            //Come back again
            timeoutId = setTimeout(increment, 50);
        })();
    }

    function stop() {
        clearTimeout(timeoutId);
        timeoutId = 0;
    }
    return {
        setTimeOffset: setTimeOffset,
        start: start,
        stop: stop
    };
}

//usage example
liveTime('current-time',-9 * 3600, 3600).start();
liveTime('current-time2', 4 * 3600 ).start();
Alaska time:&lt;span id="current-time"&gt;&lt;/span&gt;&lt;br&gt;
Moscow time:&lt;span id="current-time2"&gt;&lt;/span&gt;