Can I power up my Arduino and MotorShield with four AA batteries? can I power up my Arduino UNO with four AA batteries? It's about 6V, will my arduino burn? And, if I power my Arduino, I have to power my Motor Shield too or not?

I want to make line follower. I have to know what I have to power up. I dont know, how many battery holders I need. I have: Arduino UNO, motor shield, two 6V DC motors
You won't damage the Arduino if you remove the power to the motors as even though you're using the same power supply to power the motots and Arduino (4 x 1.5 - 6V batteries) you're not powering it from the Arduino pins itself.

What exactly are you trying to accomplish here?  Post some more info.