Bad Rim?
 

So I was truing my wheel today and I noticed no matter how much I tightened a certain spoke it wouldn't move the rim to the side. Then I realized that was the part of the rim with a dent in it, and that it was just pulling the rim downwards rather than sideways.

The rim in a non-damaged area, the zip tie is nearly in line with the edge of the rim:

http://i.imgur.com/bIFoXYS.jpg

The rim in the dented area, the zip tie is in the middle of the rim:

http://i.imgur.com/xS4hhY8.jpg

So is this rim permanently damaged? Do I need to get a new one?

Its a rear wheel on a Mountain Bike .. Its Not 'Bad' Passively
It was damaged actively.

Time for a New Wheel.

Dang. I sort of wanted to buy a new wheelset anyway though, so now I have a real reason at least.

You've got some impact damage there. By the looks of it, the brake track is (also) bent inwards.

To what degree the rim NEEDS to be replaced depends on how it feels when riding.
"If it aint broke, don't fix it".

If you haven't noticed it while riding, well, then KEEP riding.

If it bothers you, rims can (sometimes) be bent back, both into round and flat.
Personally, I've always had to unlace - at least partially - to have good results with that. But some say otherwise.
For flat spots, I pull them out with C-clamps, a plank and a crude cradle. For out of true, I either lay them like a seesaw over a board and step on them or clamp them protruding over the workbench edge and heave.

+1, you damaged the rim by hitting a sharp edge either very hard or with an underinflated tire or both. This is classic pothole damage on road bikes.

On the bright side, it's highly fixable if you have halfway decent hand skills.

The key is to get the brake track smooth so the brakes don't pulse as the shoes bind on the high spot (outward "dent"). You can do that with a hammer and support, or by squeezing it home with channlocks while using something to spread the load on the good side. Once it's close, you can finish with a file, sandpaper covered block, or simply riding it with slightly pulse braking until the shoes wear off the high spot.

I don't think the rim is worth fixing, as from your description it has both a flat spot and is physically bent in addition to the bulge. That's the reason you can't true in that spot - you can't loosen the spokes on one side - already slack, and the ones on the other side are already stretched to the limit. That's an awful lot of difficult work even for a mechanic skilled in fixing such things, and it will never be anywhere near the pre-damage state.

When truing wheels, if the wheel is not responding how you think it should to your adjustments, don't continue to crank the tension up to force it - figure out what the problem is. A strong wheel has balanced tension all the way around and simply tightening one spoke to force the rim to the side will jeopardize it.