A “Leap Second” will be added to the atomic clock on June 30, 2015

The biggest issue last time around is that there was a bug in a number of Linux kernel versions that threw things for a loop - they'd spin the CPU 100% busy, or some other issue, and that's largely what caused the issues last time - the actual addition of a second wasn't a big deal, it was that the kernel didn't cope with it.

Some applications are a little picky about seeing timestamps in ascending order, and when the leap second actually occurs, you have a one-second period where a timestamp can be older than a previous one, because system clocks will show the 23:59:59 second twice. At the millisecond level, you can have a timestamp of 23:59:59.100 that actuall occurs after, say, 23:59:59.900 if it occurs in the leap second.

In general, the leap second in and of itself isn't a huge deal, it's more that systems don't cope with it well because it's a rare event and not well tested.

The articles touches on this second being added is a problem for systems not designed to handle it. They cite Amazon, Yelp, etc.

Why is this an obstacle for systems?

I get that it would mean that your system is one second off the atomic clock, but if you sync with a time server wont't this remedy itself if the time server was updated, if it fell behind?