Hi gmiles,
I can give at least a partial answer, although it may just leave more questions.
127.0.0.1 is a localhost address, that is, it's the computer you're running on. Port 3580 is listed on http://www.iana.org/assignments/port-numbers as the National Instruments Service locator. This seems to be an internal process of LV used to track resources on the local machine (running as nisvcloc.exe). Blocking communication with it might interfere with only some components though, so your program may be unaffected.
192.168.1.1 is a local network address. I assume that LV is doing a Domain Name Server lookup on your local network to find an address. I admit to being totally in the dark about what LV might be doing here. Perhaps there's a reference in your code to another machine on the network?
Sorry I've no more details of exactly what actions are being performed. Maybe some kind NI folk will come along and enlighten both of us.
Joe Z.