One is they trap lint internally, which can often result in the internal switch that detects insertion permanently thinking there is a headphone in it (Google "phone thinks headphones are in"). The external speaker is usually muted when a jack is inserted, so when this happens, the phone doesn't make any sounds. Lint is very difficult to remove from the outside the connector.
The other problem--which the original inventors couldn't have anticipated--involves the hook switch (hangup button) on wired phone earbuds. The switch usually works by shorting the microphone to ground. The problem is that, because of the geometry of the plug, the microphone is also shorted to ground while the plug is inserted or removed. To avoid erroneously interpreting insertion/removal as a hangup, the software must wait for the button to be held for around a second. This precludes doing other interesting things with the button, like having double clicks, or clicks with different durations perform other functions. It also feels a bit awkward to have to hold it for so long.