Most, if not all of them. Unix time is leap second-ignorant and there are a number of other time sources that likewise handle leap seconds. If you do for some reason, require solar seconds or GPS time, there are other solutions for that. Most likely though, it's your platform that needs to handle the leap seconds and Linux, Mac, BSD, Solaris is more than capable of that, don't know about Windows or other esoteric systems though.
The issue is in human date/time representations and/or bad programming. I've seen many times programmers parsing a MM-DD-YYYY HH:mm:ss string rather than a proper timestamp. I've also seen programmers implement timers even in lower-level languages like C by simply subtracting "$current time" - "$past time" or having a timestamp with a "unique" requirement. If you've ever worked on platforms that do not keep accurate track of time or have no hardware clock such as embedded devices, virtual servers etc, then you know that it's a bad idea.
If you're relying on mm:ss to progress 'properly' during these time events, then you're in trouble. The article gives the example of timers as well which just gives me the heebie-jeebies.