Wrong target voltage in Li-Po charger

What would be the consequences of using a charger that targets 4.3V to charge a 4.2V Li-Po battery?

Will the battery be damaged? In what way?

You will shorten your battery's life or, worse, damage it. Ultimately it depends on the characteristics of the cell you are going to charge. You should browse the datasheet of the cell to get the optimal target voltage.

Quoting from the TI application note snva557 (emphasis mine):

LI-ION CHARGING INFORMATION

A Li-Ion battery is unique, as it is charged from a fixed voltage source that is current limited (this is usually referred to as constant voltage charging).

Constant Voltage Charging

A constant voltage (C-V) charger sources current into the battery in an attempt to force the battery voltage up to a preset value (usually referred to as the set-point voltage or set voltage).

Once this voltage is reached, the charger will source only enough current to hold the voltage of the battery at this constant voltage (hence, the reason it is called constant voltage charging).

At present, the major Li-Ion cell manufacturer recommends 4.200 +/- 50 mV as the ideal set point voltage, and 1c (a charging current rate equal to the A-hr rating of the cell) as the maximum charging current that can be used.

The accuracy on the set point voltage is critical: if this voltage is too high, the number of charge cycles the battery can complete is reduced (shortened battery life). If the voltage is too low, the cell will not be fully charged.

Note the very tight tolerance on the target voltage: 4.200V ± 50mV, that is a range from 4.150V to 4.250V. Your 4.3V is 50mV larger, assuming it is 4.300V, i.e. precise to the 3rd decimal digit. That's critical: when you say 4.3V what do you mean exactly? If it is 4.300V (guaranteed maximum) it is 50mV higher (which is bad). But if it is 4.3xxV, where xx are unspecified digits, it could be also 4.390 (say), and that would be 140mV too much!

And, yes, in the long run it could also cause swelling or worse things, if your battery is not protected by a protection chip.