I have three one-way tickets for a multi-destination trip over a 3-week period. I am flying the first and the last flights with the same airline. I want to skip my second flight. Will my last flight be canceled?
The answer depends on what you mean by "ticket."
If you have three separate and independent tickets - you purchased each ticket from a separate and different airline at a different time, or even from the same airline but with each flight having a unique record locator number - then the air passages will be independent. You can skip any flight(s) without affecting your ability to take any other flight(s).
If, on the other hand, you bought all three flights from one airline and all three flights carry the same record locator number, then all three flights are "connected" in that airline's records. As Giacomo Catenazzi points out, airlines' terms of carriage often or usually provide that if you don't show up to take one flight segment, then any future segment(s) will be cancelled.
Without knowing more about the form of your tickets (as dda mentions), and the terms of the airline(s) terms of carriage, it's impossible to answer the question.
If you booked them on a single ticket, chances are (dependent on the conditions of carriage, which vary by airline) that your final leg will be cancelled when you miss the middle leg.
If you booked them as three separate tickets, your last booking will not be affected by missing the middle booking.
If you no-show for a flight then all remaining flights on that ticket will normally be cancelled without refund.
So the question becomes do you really have three independent one-way tickets or do you have three flights on the same ticket.
If in doubt your best bet is probablly to contact the airline or agent you booked the ticket(s) through. They should be able to figure out if the flights are on the same ticket and if-so they should be able to change the ticket to remove the second flight. Depending on the details of the ticket this may cost you money, it may be free or it may even result in a refund.