[NineRedTools]

I was listening to my DT770s at a louder-than-normal volume (albeit nothing extreme), and I started noticing a quieter sound in the left ear as well as buzzing. After opening the headphone up, I see this:
 
There's a small tear in the side. I don't know how it got there. I know driver replacements are ~$50, but I want to know if there's any way to repair that paper membrane thing or replace it in a low-cost way

 

[billybob_jcv]

I'm pretty sure that is part of the actual driver. Isn't there supposed to be a black piece of fabric or foam covering that white paper?

 

[cskippy]

i made a small puncture like that when i was modding my DT880. I don't notice any difference. Make sure that the drivers are set properly. They should only go in wben they line up with a slot.

 

[NineRedTools]

I'm pretty sure that is part of the actual driver. Isn't there supposed to be a black piece of fabric or foam covering that white paper?

There is a black foam covering. But the paper isn't a part of the driver itself. It's just a covering.

After putting everything back together again, things are working just fine now. Odd.

 

[GREQ]

 

I'm pretty sure that is part of the actual driver. Isn't there supposed to be a black piece of fabric or foam covering that white paper?

Not quite.
This is a fabric/paper like material that behaves as a 'bass-trap' and dust cover. I think it's also responsible for reducing sound reflections from the cup, and 'increasing' soundstage etc etc etc
 
Anyway... the point is, that is NOT the driver. The driver is the same size as and sits behind the grey disk in the middle.
 
 

After putting everything back together again, things are working just fine now. Odd.

It might be working now, but that's probably because the loose bit has become 'stuck'.
You should consider fixing it since if it CAN happen again, it will.
 
I would get a TINY amount of PVA glue (or wood glue) like the size of a lentil, mix it with a tiny drop of water with a small paintbrush, and gently apply a TINY amount of the glue on both sides of the tear.
Since both sides are a little wet with glue, it should be easy to get them to stick together and hold.
Use a sewing needle (or similar) to get it in the right place... just take your time until its properly aligned.
Then let it try for 24 hours (this is the hardest part 

) so the tear is in a standing/vertical position. (if you leave it flat, the weight of the glue might cause it to separate while it glues).