Input voltage from circuit drops despite Arduino turned off
To protect the internals of integrated circuits many contain diodes at most input and output pins to VCC.  In normal operation where the voltage at these pins are not expected to become greater than VCC, these diodes are not in forward bias.  However, should the unexpected happen and the voltage at these pins does become higher than VCC the diodes are in forward bias. 
 That is to say, the diodes try to keep the voltage at the pin no higher than VCC.  

When you removed the power from the Arduino you essentially lowered VCC to zero.  This will likely forward biased the diodes inside the Arduino and drive the voltage at the pin toward zero.

There is a good discussion of this in this Electrical Engineering StackExchange question &amp; answer.