I am curious if these leap seconds make their way to my computer in any way?
From the JavaScript date object, I can see that they are not included in any way. The base representation appears not to include them at all.
It appears they are added when the clock synches. How does this effect JavaScript time?
Following the comments below I adjusted the time strings in my "play around" snippet to GMT (with a "Z" at then end).
console.log(new Date("2015-07-01T00:00:00Z")-new Date("2015-06-30T23:59:59Z"));