I'll limit my answer/addition to "burn" type failures, because otherwise there's really a mile-long list of circuits that can malfunction (any high-pass filter, any feedback loop etc.) and depending what they control stuff might burn/blow up. The example that SunnyBoyNY gave with the SMPS output cap is actually an example of feedback loop. More modestly, any opamp can (and will) oscillate with a big/right capacitance on its output; and when it does so it will heat up. Unless it has thermal protection/shutdown, it may get damaged. In all these cases it's the reduction in phase margin by the [increased] cap that's causing a problem. I won't get into details on this, because LTI control theory is almost certainly too difficult to grasp [right away] by someone wondering about this too-big-capacitor-damage issue.
However, even in a linear power supply, a capacitor that's oversized relative to the rectifier (diode) can lead to failure of the rectifier via increased peak [charging] current. Below is a textbook illustration of the problem (for a half-wave rectifier with capacitive filter):
The smoother the filtered voltage gets (less ripple), the smoother the load current gets, but the same energy is transferred to the capacitor over a shorter period of time (conduction interval) so the diode sees a higher repetitive peak/surge current when the capacitor is increased. Actually this is just a repetitive version of the worst-case scenario, which is the inrush current when the capacitor is completely discharged (in which case the peak will actually be much higher), but it's worth noting that diodes have different/lower current ratings for repetitive stress vs "non-repetitive" events (which is a bit of a misnomer, such events can repeat just nowhere near the mains' frequency; this appnote is good read for diode-spec jargon.)
A less obvious variation [because it doesn't involve things sold as capacitors] of this inrush current issue is the gate capacitance of a MOSFET. Make it too big [say, by changing/upping the MOSFET model so it can switch a bigger load] and the gate inrush current can damage an IO pin of a microcontroller...; avoiding such a scenario is one of the [many] reasons why MOSFET driver circuitry can get complicated.
Finally, the problems mentioned are not insurmountable. One can compensate feedback loops (one way of doing that is actually by adding/increasing capacitor[s], but in a different part of the circuit). As for limiting inrush current (directly) there are numerous solutions, the most obvious being adding an inductor.