Q:

Will there still be small voltage in transistor OFF state?

In Bipolar junction NPN transistor which has three legs as shown below:
 
Case 1: If this transistor is in OFF state i.e. collector is connected to +Vcc and Base is connected to insufficient voltage which is less than the required amount to turn on the transistor, would it still conduct small amount of current (or low voltage which is not zero)?
Case 2: In other case, if only Base is connected to +Vcc, would there be some current in collector or emitter?
What am I trying to know is that does the OFF state in transistor mean zero voltage at collector and emitter?

A:

First, you mix voltage and current. Voltage is the "pressure" on electrical charges, so they want to move. Current is the flow of electrical charges, if the find a way.
Case 1:
To answer the question, have a look into this datasheet of a common BC337
The table on page 2 states a Collector Cutoff Current of 100nA for V_CE=45V and V_BE=0V.
So yes, there will be a current through the collector. Keep in mind that a NPN transistor looks like two diodes connected back to back:

simulate this circuit – Schematic created using CircuitLab
While this doesn't describe a transistor in general, here it does. You are operating the upper diode in reverse direction. In this case, real diodes will show a small leakage current, 100nA is a perfect sample for it.
You didn't say where E should be connected to, but if it's connected to GND, as in my schematic, the lower diode is shorted by the connection between B and GND, so the 100nA will flow through the base leg. It's just the same as if E is not connected anywhere. If E is also connected to Vcc, you just get a second diode in parallel to the first, so about twice the current (depending of the characteristics of the diodes)
Case 2
Well, if only B is connected to Vcc, and the rest is not connected to anything, no current will flow. If you connect E and/or C to GND, the diodes will be operated in forward direction, and a significant current will flow (and destroy the transistor if currents are niot limited by resistors)

