What is the effect on kWh if the power company reduces line voltage to a residence?
We just received our power bill and it is showing a 30% rise in KW for the past month. 


Check your bill. It will be showing "kWh" which is energy usage, not "KW" (which actual signifies kelvin-watts - you meant "kW") which is power, the instantaneous or average power reading.


  Nothing has changed in our home. 


This gives us a few options:


You are mistaken and your usage has gone up. Take a daily meter reading and see what's going on. Compare this with the daily average calculated from your bill.
The meter is faulty.
The billing periods or meter readings are inconsistent. e.g., 25 days one month and 35 the next.
Some of the bills have been using estimated readings. These should be indicated on the bill. If you have had several low estimated bills then the next meter-based bill is going to come as a surprise!



  This past summer our stove would not ignite because the line voltage was 108 V.


In general, reduced voltage will result in reduced current. Since power is volts x amps the reduced voltage will result in reduced power too. Exceptions to this will be for devices such as heaters or air conditioning which will remain running until the setpoint is reached. With reduced power available this will take longer and, if they run inefficiently at reduced power then the total energy used will increase.


  ... the electrical supplier and they came out, confirmed the condition and changed the tap ... They then came back out weeks later, but I don't know what they did. (Changed it back, perhaps?) 


A cheap multimeter will allow you to periodically check the mains voltage and see what effect your load is having on it.



First thing to do is take your daily meter readings to figure out what's going on.