Will mining significantly increase the power consumption of my PC? I have a PC that is always on 24/7. I use it as an SSH and backup server, and it has a modest, not-too-great GPU that sits idle most of the time.

I hear a lot of talk about mining not being cost-effective for most people because electricity isn't free. However I'm already paying the electricity bill for my PC. I might as well let it sit and mine all day while it's on, right? Maybe help pay the electric bill?

Or will mining increase my PC's power consumption so much that it's still not worth it?
Yes it will. The component you're likely to use for mining is your video card (GPU) which consumes very little power when idle but can consume a great deal under load. Check this list of common cards and consult the Watts (W) column for how many watts your card is likely to draw. Any card worth mining on will probably draw at least 100 Watts under full load. Depending on the electricity costs in your area, it may cost you more in electricity than your coins produced are worth. There is at least one good calculator to estimate your proceeds but you'll need to find out how much your local utility company charges per kilowatt-hour for electricity to determine if it's profitable. For reference Kilowatt-hours per day = Watts * 24 / 1000.