Electrical – Can induction in parallel ungrounded cables lead to false continuity readings with a multimeter

I'm guessing the circuits are sharing a neutral, and so are considered a multiwire branch circuit.

Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system.

In the 2008 version of the National Electrical Code (NEC), they introduced sub section (b) to section 210.4. This sub section covers the disconnecting means of a multiwire branch circuit, and requires a simultaneous disconnect of all ungrounded conductors.

NEC 2008
210.4 Multiwire Branch Circuits.
(B) Disconnecting Means. Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates.

This subsection can be satisfied by using a multipole breaker, which provides both simultaneous trip and simultaneous disconnect. The subsection can also be satisfied by simply attaching handle ties between multiple single pole breakers, which would provide simultaneous disconnect (and may or may not provide simultaneous trip).

What you are describing is called a Multi-wire Branch Circuit, and it is completely legal (given that it's done properly).

The first problem, is that it doesn't should like the breakers feeding this circuit are handle tied together. It should be feed by a double pole breaker, or the two breakers handles should be tied together. You should not be able to shut off only half the circuit (red only or black only).

As for the voltage. I'd suspect it's phantom voltage (inductive coupling), and so should not cause any problems (especially if the breakers were properly tied together).

If the breaker for the "red" conductor is off, and current is flowing on the "black" conductor. Because the conductors are parallel, and in close proximity, a voltage may be induced on the "red" wire. There should be no current, however. So if a load was introduced, the voltage would drop to 0.

You can determine if it really is phantom voltage, simply by using a low impedance voltmeter. The low impedance meter will read 0 volts, while a high impedance meter will read the phantom voltage. An analog meter is one example of a low impedance meter, though some digital multimeters are able to operate in both modes.