This is a very interesting topic and will be worth following as it unfolds. Let's see...
A) Rise because many people are fleeing to Bitcoin as a store of value, such as with Cyprus and China. or
Assuming the net still works, very likely. And it most probably will since so much of our infrastructure depends on it.
B) Fall in value due to a REDUCED demand due to:
1. People exiting bitcoin for more reliable stores of value such as gold.
Gold is only more reliable if the net goes down. Otherwise it is less reliable as it can be confiscated, which might very well happen.
2. Bitcoin primarily does business in the U.S. and primary trades against USD, which will both be crippled.
Why would this matter? Setting the US aside, other countries will fear for their economic lives if the dollar implodes so they will be looking for some kind of way to get through the inevitable following shitstorm safely.
3. There will be no interest in speculative investments once people enter into survival mode.
Bitcoin is economic survival in a world of inflation. Moreso hyperinflation.
4. The banking system may be disabled, leaving people unable to send money to exchanges.
Localbitcoins. Better yet, just trade with your neighbors. Eventually we may have regular bitcoin meetups in local neighborhoods all over the world.
5. The central banks may ban business with bitcoin exchanges such as in China or Russia.
Prohibition worked so well. Doesn't matter, unless they can take the blockchain itself down (they can't) bitcoin will be fine.
6. The internet will be inaccessible.
This one is legit.