This fear is very similar to the overblown Y2K crisis. Many feared that computers without the software capabilities of handling years above 1999 would lead to technical failures across all sectors of business and society, however, little to no actual disruption occured. Considering modern computers no longer face similar hardware limits (as was the cause of the Y2K problem) there should be no trouble adapting to a one second difference.
Most computer programs have date calculations based on days, not hours and seconds. The end user won't see anything. But the strongest evidence of this is the leap second added in June, 2012. Did that leap second break the Internet? No it did not.One of the biggest historical issues in this category of computer-related problems was the Y2K debacle which was supposed to bring every computer to a complete halt. Remember that? That was a complete snooze. This will be too.
The leap second is resulting in unfounded worry, much like the concern for Y2K. Computer programs are more sophisticated than to be stymied by a simple time issues - and those that are have been accounted for and modified. This is an example of uneducated panic and fear mongering by those who don't fully understand the issue.
No, I do not think that this years "lead second" will have dire effects on the internet because there has been talk about this going on for a long time and it hasn't effected anything up until now so I can hardly see it affecting anything really. I guess only time will tell.