If your "test" was fast, then multithreading would not have given you any advantage whatsoever, because your code would be 100% disk-bound, and presumably you have all of your files on the same disk: you cannot improve the throughput of a single disk with multithreading.
But since your "test" will be waiting for a response from a webserver, this means that the test is going to be slow, so there is plenty of room for improvement by multithreading. Basically, the number of threads you need depends on how many requests the webserver can be servicing simultaneously without degrading the performance of the webserver. This number might still be low, so you might end up not gaining anything, but at least you can try.
If your file is not really huge, then you can read it all at once, and write it all at once. If each line is only 80 characters long, then this means that your file is only 8 megabytes, which is peanuts, so you can read all the lines into a list, work on the list, produce another list, and in the end write out the entire list.
This will allow you to create a structure, say, MyLine which contains the index of each line and the text of each line, so that you can sort all lines before writing them, so that you do not have to worry about out-of-order responses from the server.
Then, what you need to do is use a bounding blocking queue like BlockingCollection as @Paul suggested.
BlockingCollection accepts as a constructor parameter its maximum capacity. This means that once its maximum capacity has been reached, any further attempts to add to it are blocked (the caller sits there waiting) until some items are removed. So, if you want to have up to 10 simultaneously pending requests, you would construct it as follows:
var sourceCollection = new BlockingCollection<MyLine>(10);
Your main thread will be stuffing sourceCollection with MyLine objects, and you will have 10 threads which block waiting to read MyLines from the collection. Each thread sends a request to the server, waits for a response, saves the result into a thread-safe resultCollection, and attempts to fetch the next item from sourceCollection.
Instead of using multiple threads you could instead use the async features of C#, but I am not terribly familiar with them, so I cannot advice you on precisely how you would do that.
In the end, copy the contents of resultCollection into a List, sort the list, and write it to the output file. (The copy into a separate List is probably a good idea because sorting the thread-safe resultCollection will probably be much slower than sorting a non-thread-safe List. I said probably.)