but I'm pretty sure the pins only communicate data, not power? Do I need a shield which can regulate the power to each of these circuits to control when they receive power?
Right. The I/O pins Arduino put-out 5V at 40mA or less.
How much current do you need? Or, if you know (or yiou can measure) the resistance you can calculate it using [u]Ohm's Law[/u].
You can build driver circuits for each separately-controlled thread or you may be able to use a multiple-driver chip.
[u]Here[/u] is a MOSFET motor driver circuit. In this circuit your threads would share a common power supply and they are powered/controlled through the negative "ground" side. (With a resistive load you can eliminate the diode.)
The open power line is sewn into a piece of fabric covered with thermochromatic ink, which responds as the thread heats up.
Have you tried this without the Arduino? It seems like the heat will be concentrated around the thread and it might give a "small" effect. And, that's assuming you can get enough heat to get temporary color-effect but not enough heat to get a "fire effect". :D
to open power lines (conductive thread).
In electronics terminology, "open" means no-connection. For example, if you open the connection to a light bulb, no current can flow and the light will go off. Although sometimes the connection should be open, such as when a switch is turned-off, when somebody says "open" they are usually referring to a fault.
The opposite of an open is a "short". i.e. If you short the two wires to the light bulb there is no resistance through the bulb, excess current flows and you blow a circuit breaker. Again, most of the time when people say "short" they are usually referring to a fault, but sometimes someone will say, "You forgot to short the grounds together".