Can you share some pictures?

Can you tell us a little about the potting medium?

How often do you water?

What kind (composition) of citrus fertilizer did you use?

How big is your container?

As long as there's some green there's hope - but I do think you are going to need to move quickly.

I believe the potting mix was Sta-Green Potting Mix.

I have watered as little as twice a week and as much as every couple days. There were times it had a number of leaves, then I'd water and the leaves would all fall off. I thought maybe the soil was getting too dry because I read that if the soil dries out and the tree is then watered, it will dump its leaves. I've been fighting with leaf dumping a lot. It will look like it's doing well, lots of leaves and flowers, and then everything falls off.

Fertilizer was Miracle Grow Shake 'n Feed 13-7-13.

The container is 14.5"w x 15"h

I have included a link to some photos of the tree I took tonight. I can take more in the morning. I also included two pictures from when it was very leafy in its old pot (our cat was only sniffing, not eating), and one with the aforementioned sunburn-looking spots.

Thanks.

Here is a link that might be useful: Lime tree photos

I know you'd like to help this poor thing survive and grow.
But since you're so new to the process, how about eliminating a lots of sleepless nights and replace it with a fresh plant.

Ask lots of questions before you take the newbie home. Perhaps they have an info sheet for you.

To start, know that potted plants can cook rather rapidly in summer's sun, especially in TX.

Hacktrack, you might not consider my advices/ideas as I might be even less experienced than you, but what I would have done in your case is to have a water tray filled with stones/perlite and water. From my own experience this is how I killed my coffee plant that I thought is easy to grow at home and would harden pretty fast. Now I know that lots of exotic plants does not love the strong sun but more like an afternoon one + lots of humidity. Now I have a similar problem with one Eureka Seedling I am sure have rooted as a few tiny leaves have grown but it just stopped growing and nothing visual appears (at least not over the soil ;) ) so what I plan is to have a large tray on my window that receives a lot of sun and to fill it with water so to simulate a natural humidifier and see how it would go from there.

My Buddha's hand citrus have also stopped growing but the leaves are perfectly strong and green so I guess it is also lacking some humidity to reduce the stress. It might be also a reason that I have changed the pot to a larger one so now most of the energy is focussing on the roots but not on the leaves growth etc. but would add the water tray and see it from there.

One thing to bear in mind also is that the plants need some food when grown in pots but overfeeding might burn them, so if you have added some fertilizers, try to get an organic one with a slower release and/or at least lower NPK so to try to feed weekly and see whether that would help. I have at home General Organics liquid plant food which is 4-3-3 with some minerals and sea weed mixture for minerals. I guess we all have to experiment, but if you grow the citrus on a balcony or outside, try to put it somewhere more shady and leave it to pass the critical moments. If I was on your place, I would not just give up the fight and might even try to clone a few cuttings so to keep at least the plant (if that grows ;) )

Thanks everyone for the tips so far.

I have repotted the tree and moved it into more direct sunlight. When I attempted to lift the tree out of the pot, it came right up with no effort needed. There were lots of roots, but they didn't seem to go anywhere. The soil at the bottom of the pot was a big mess of mud.

I removed all the soil from the pot and washed it out, washed the soil off from around the roots of the tree, and repotted it in the Miracle-Gro Cactus, Palm & Citrus potting mix and mixed in some perlite and bark. There's also a layer of gravel on the bottom of the pot.

I have moved it outside of the covered patio into an area where it will get at least 12 hours of sunlight. It is also near the lawn, so two nights a week it will catch some mist off of the sprinkler system.

We'll see if this leads to any signs of recovery.

Many people will tell you that gravel at the bottom will do more to clog up things than to promote draining. http://forums2.gardenweb.com/forums/load/contain/msg0223050125125.html?8276