I didn't drink it, but wonder if it could affect me or my stomach.

If you didn't drink it, where's the harm?

I'm guessing, had you drunk the tea, there would be enough liquid to offset any germs plus your stomach's gastric juices would have taken care of any microbes.

Drain flies spread disease with their bites.

Do you live in the tropics?

No, I actually live in an urban area. There's been drain flies in the apartment since I moved in. I'm trying to manage them, but occasionally they get into the refrigerator and into my food.

Be sure to carefully cover your food and put drinks into sealed containers/pitchers when storing them in the refrigerator.

Besides the aspect that drinking fly juice is mostly just gross, if you know that you are looking at a drain fly, you can take comfort that you cannot be parasitized by drain flies, and drain fly juice is not toxic. The group of flies known as drain flies live in, guess what? drains. The larvae feed on the slime within drains and on damp surfaces. I suspect that if you are finding them in the refrigerator, they may be surviving on the slime growing in the condensation inside your fridge and are sneaking into it through the ventilation.

Drain flies do not bite and are not known to transmit any disease.

There is a somewhat closely related fly, the sand fly, that transmits, through biting, a nasty disease called leishmaniasis (http://www.ncbi.nlm.nih.gov/pubmedhealth/PMH0002362/). Luckily, for the urban dweller, the fly is primarily found in tropical habitats and rarely in the developed world. Wikipedia has a good description. (http://en.wikipedia.org/wiki/Leishmaniasis). I found this fun fact: "Leishmaniasis (http://homepages.uel.ac.uk/d.p.humber/akhter/trans.htm) is primarily a zoonotic disease in which wild and domestic animals such as the fox, jackal, rodents and wolves serve as reservoir hosts. Other animals in the surrounding areas can become infected and these are referred to as secondary or incidental hosts. Of all the potential animal hosts, domestic dogs by far play the most important role in harbouring and transmitting the disease to humans due to the close association between humans and dogs as pets."

Get rid of the dampness, get rid of the slime and the drain fly should become less of a problem. Good control methods can be found at:
http://www.ca.uky.edu/entomology/entfacts/ef615.asp
http://www.pestid.msu.edu/InsectsArthropods/Mothfliesordrainflies/tabid/283/Default.aspx

What all disease can I get from drain flies