What do I do as a depressed and incompetent TA?
I utterly disagree with Mad Jack. You do not need to resign and by resigning, you set yourself up for a future where you will think of resigning every time you have a task that you do not think you will be able to complete to the high standards you set for yourself.
Try to disaggregate the two things -- the depression and the incompetence. If you are merely incompetent (or not as competent as you would like to be), read the rest of this answer. If your depression is putting you at risk to yourself or others (and will not be alleviated by acceptance that one cannot be competent at all things all the time), then perhaps Mad Jack's advice is warranted.
Now, about incompetence.
The reality is that you cannot be good at everything you do. Not all classes go well - even with the best of intentions. Sometimes, a class just fails. Sometimes it's your fault, sometimes it's the students, sometimes it's a combination. There are many classes that I've wished that I could have quit teaching halfway through.1
Sure, after you finish this semester your TA evals might be bad, they might be scorchingly bad. But that's ok. You fulfilled your obligation -- poorly, perhaps -- but you fulfilled them. The students themselves will recover. It's terrible that we can't all be always be stellar teachers, but we're human beings. You're just a blip -- one section of one of the 32 classes they take at the university.
You've found out that teaching does not come naturally and that teaching is hard. This is a good life lesson. It's part of your graduate education. You will get better at teaching the more you do it -- or at least figure out how to work around your incompetencies. We all do.
1. In the interest of full disclosure, I also struggle from depression and so this is written to you from the position of a peer. This advice is of course null and void if you at risk of entirely melting down. Your first duty is to yourself. But only you can know how close you are to endangering your own health for the sake of the class -- not anonymous internet peers.
Anecdote: I was forced to TA a class on a topic that I knew nothing about when I was a grad student. It quickly became clear that the even the most basic students knew infinitely more than I did about the subject. I was a terrible, terrible TA but my professor worked around me, the students learned to never come to my sections or office hours, and somehow we made it work. Now every time I get assigned a terrible TA, I believe it to be karmic retribution for my past sins. So talk to your professor. S/he may have also been a terrible TA and may be more than willing to work with you on this.
Finally, mood disorders like depression and bipolar disorder are not incompatible with academia. There are many successful academics with tenure (Kay Redfield Jamison and Emily Martin have both written books on their own experiences with bipolar disorder). Yes, during our trough periods things can be hard to do, but when the fog lifts we tend to be more productive. I've used that successfully to write two books, publish numerous articles, blah blah. Fortunately I'm at a university that allows for some umm ... eccentrism in the faculty body. I am three sigmas on that scale. :-)
Ohhh, no, I'm so sorry you're going through this. I deal with depression too.
First, you really need to recognize that everything you're perceiving, you're perceiving through a Fog of Depression. Your brain is attacking your self-esteem and motivation, and also causing you to worry about things that you don't need to worry about nearly as much as you think you do, and it seems even more awful because you can't see an end to your problems. There is, in fact, an end--there is a solution out there--and you can find it and your life will get better again.
This includes your perception of your past performance. You have been doing better than you think. (Yes, you have.) Don't dismiss your efforts.
Second, talk to your professor. Straightforward honesty can help you get to the root of your problems and start fixing them. If your prof is a human and not a jackal in a human suit, they shouldn't think any less of you for something you can't control. (No, you can't control the fact that you have an illness. You're not just being lazy.) There may be school policies in place protecting you from discrimination based on mental illness, which may or may not be a thing you'll see. Your worries on this count may be entirely baseless depending on what the people around you are like. Remember, you are chemically disposed towards pessimism right now and it's screwing with your head--you need to learn to recognize the thought patterns that cue you in as to when it's the depression talking.
If you don't tell your professor, then at least tell SOMEONE about your diagnosis. It doesn't even have to be someone official. Just tell someone who won't make you feel like crap about it and will maybe go to Dairy Queen with you sometimes to cheer you up. Getting out of the house/apartment/dorm/whatever is really important.
Third--It's the middle of winter; could your mental illness be worse because you've got some seasonal affective disorder on the side? If so, you should buy some full-spectrum "daylight" light bulbs. I'm talking about the kind where the color temperature is something like 5000 or 6500, instead of 2700 or whatever lower number. This helps replace sunlight in winter when your brain is missing it, and using those light bulbs in your living spaces, like in your desk lamp and so on, can help dramatically with SAD. I know it sounds like one of those bizarre homeopathic remedies, but it does work--there are studies out there and my personal experience backs them up, you can Google for them if you're interested.
Fourth, sometimes you do need to take time for yourself. Ask yourself: would the mental health professionals at your university really recommend something that would ruin your career? Where did you get the information that taking time off would burn bridges? Is it based in fact, or in anxiety? I'm not saying that your worries are for-sure baseless, but you need to double-check. You can tell the counselors about your concerns, too. And maybe you don't need to take a lot of time off--maybe a month while you try out a medication, or even less time while you just take time out to relax and take care of yourself.
I know I have a bad habit of taking things on that I can handle academically--like, I learn well enough to take them on, they're nothing I can't understand--but they're too difficult to keep up with emotionally and I have to step back one way or another. That's okay. Everyone has limits, no one's Superman, you're not a robot. But because you're not a robot, your limits WILL change over time. You've learned you're dealing with a problem; once you adjust to coping with it and find out what works, your limits will expand again. Sometimes you need time away from stress to get to that point though.
Fifth and related: take time out for self-care. Bake a cake (like this one), take care of a houseplant (green oxalis or african violets will bloom indoors), learn how to properly keep a betta fish (the people over at the forum bettafish will happily tell you how to make a betta live for years instead of months), cook yourself dinner instead of ordering takeout, read a few chapters of a kids' fantasy book each night, take a long bath with soap that smells nice, spend time with your friends, listen to music, leave the house just to buy a cookie at a coffee shop, follow a web comic (you might like this one--hold out on it, the art starting in Vol2 is pretty odd but it gets better fast).
This point might seem off-topic, but it isn't. You really do need to make sure that the thing that's been stressing you out isn't your whole life.
Good luck :) We're rooting for you!
There are at least two issues, what to do about your TA performance, and what to do about your depression. You need help with both.
Depression could affect your self-evaluation. Your performance as a TA may be fine with some minor problems, or could be really bad, and you might not be able to tell the difference. You need an objective observer to evaluate how you are doing before deciding what, if anything, to do about it.
Similarly, if depression is affecting you to the point where you at least feel you are incompetent, whether accurately or not, you may need medical help with that.
The real problem is your feeling that seeking out help will risk "irreparably burning bridges". I know by practical experience that graduate students needing leave or other accommodations for medical issues is something that happens, something for which colleges have procedures in place. I don't know why you feel seeking help from the school counseling center, and if necessary taking medical leave, would be such a serious step in your case.