To call 'under voltage' a "danger" would be a bit misleading. Seldom does operating a piece of equipment at below suggested operating voltage cause it damage, except in the case of motors (especially if they stall) and where it causes two circuits (or portions of the same circuit) to operate against each other. This last situation is covered better in "latch-ups" in the section "equipment failure".
Generally, the issue with under voltage is there is not enough supply to "get things going", again the motor is a good example and it is not uncommon to find extremely lengthy high start currents until the motor finally starts turning.
Lighting circuits are also sensitive to low start voltages, especially those that use simple "starters". The current through the lamp is not sufficient to halt the starting circuit and the lamp is forced to endure continual starts. A prime example is a fluorescent lamp that flashes all day long. Many times it is a faulty component such as a 'starter' or the lamp itself, but many times this failure is from bad power.
Electronic circuits too can suffer "long start ups", or just not start at all.
There are two aspects to under voltage with regards PSUs, and they are divided by the technology used.
Older linear regulators are subject to a minimum operating voltage and if below that there was no longer any "headroom" for the regulator to do its job. This headroom had to also allow for ripple on the smoothing capacitors. Many older pieces of equipment were fitted with voltage selectors allowing changes of approximately 10%. Good pieces of equipment had at least 200, 220, & 240 slots. Better pieces could do down to 5V increments.
Most circuitry can accept (said with reservation) when the voltage falls below that where the linear regulator can maintain regulation. A Hi-Fi would be a good example with the speakers giving off a distinctive hum with the circuitry towards the input not being fed with a stable voltage. If the circuitry were to be sensitive to this e.g. the biasing being upset causing high current drains, then the low voltage condition could very well lead to failure in the parts being put under this strain.
Although this same condition can exist in SMPSs, it is far less pronounced and is usually well below any possible operating voltage (typical cutoffs are at least 40% of operating, and as low as 85VAC in 110..240V input capable units). A SMPS generates some very interesting properties when operated at below its cut-off.
The danger with SMPSs and low voltage is they will want to draw more current in order to deliver their output. More current will mean a higher drain on the mains, which will mean more current draw.... and the loop is perpetuated.
This is not more visible than when operating IT equipment on reduced voltage - and one which has got many data centres in the USA to consider running their servers etc. on either 208 or 230V as opposed to their standard 115V.
Most IT equipment (if not all by now) uses 'universal input' (100-250V) switch-mode power supplies. The primary component requiring focus here is the storage capacitor following the rectifiers. For the sake of simplicity, efficiencies and other factors are not going to be incorporated in this explanation.
A 230W power supply will require 2Arms at 115V, and 1Arms at 230V. The storage capacitor will be delivering such a current almost continually (except during the portions of the cycle when it is being "topped up" via the rectifiers).
From personally conducted tests, SMPSs die at about 60VDC or 45VAC - some are lower, some higher, this is an average (and also very dependent on hard the power supply is working at the time).
During a dip at 115V, the capacitor will need to supply a full 2A of current to the rest of the SMPS, and will only be able to supply such a current (which will be ever increasing as the voltage falls) till the lower operating limit is reached of about 60V when the SMPS stops supplying power to the server/PC/etc. Before the dip occurred, the headroom was approx 115x1.414 - 60 = 100V
At 230V, the capacitor only has to supply 1A, and has a massive headroom of 260V (i.e. the amount the voltage needs to fall by before the SMPS quits supplying output). Not only is the headroom larger, but the current draw from the cap is half therefore the discharge rate is half that of when a dip occurs at 115V.
Using very simple maths, it is not too difficult to imagine the immunity to be increased by a factor of
((260/100)*(2/1) = 5
i.e. the dip could last about 5 times longer before the PSU quits on the device it is supplying as opposed to when run on 115V. No prises for guessing which voltage I prefer my PC being supplied on, especially as the incoming supply can suffer anomalies!
In some domestic equipment, which has a narrower voltage operating range, the higher current because of a lower operating voltage may start to strain the switching transistor either through the heat the insufficiently cooled transistor needs to dissapate, or simply the current approaches the maximum for which the transistor was designed. Either (or both) of these factors can drastically reduce the working life of the part from many years to merely a few weeks, or months at best.