Disaster!

No you don't have to cut it out.  What kind of extensions are they?  Call her up and ask what they are.  They are either going to be shrinkies and will come out by applying heat to the shrinkie to slide it off and then using a lot of oil on the glue, or fusion, in which case it will either be wax based and come out with a lot of oil, or be keratin based and come out by applying acetone and pinching the glue with a pair of pliers. That will turn the glue crumbly and powdery and you can brush it right out.  I'd give her a call and ask her what she used on your hair and the brand name. 

Just out of curiosity, what do you mean by "leaves a line of glue residue"?  Do you mean that there is literally glue down the length of your hair or is a small generalized line closer to the scalp?  Also, what do you mean there's glue everywhere?  Is it literally all over your hair or do you mean that there are a lot of "spots" of glue that are close to the scalp?  Sorry for the questions.  I'm just curious about the details. 

"Line of glue" sounds like latex glue-bonded wefts to me. In which case, it will come out with oil. Soak the gooey spot of hair with conditioner, then omb out and wash your hair. And try to keep comditinoer off the rest of the roots.

And yeah, 2 weeks is not an impressive lifespan, even for latex glue. I'd mention that to the stylist.

-Rae

Rae, also thinking the same here.  Sounds like latex, but she said that every week 2 fall out, so if they've been in for a month, that would be 8 wefts, which doesn't sound right.   So that  lead me to believe they were strand by strand.  But you wouldn't use latex for strand by strand.  I'm curious for more info.