Will current flow through my digital multimeter while it is powered off?
In short, yes.

The way a digital multimeter measures current (not a loop current meter) is that it measures the voltage drop across a small precision resistor. For reliability, simplicity, and repeatability, no switches or contacts are placed in series with this shunt resistor. Some meters will have a fast-acting fuse to protect the shunt resistor. Basically, the switches will control the electronics hooked up to the shunt resistor, but they won't disconnect it.

You can perform an experiment with your meter as some degree of proof that this is true. If you configure your meter to read resistance, put the positive probe into the positive current probe socket. You should measure a very small resistance. This shows that the meter doesn't disconnect the shunt resistor while in a separate mode. It would be a small leap of logic to say that if it didn't switch off the shunt resistor then, it probably couldn't when the meter is off. 

Alternatively, you could measure the current using a second meter.