Electronic circuit switch for arduino
It's hard to tell from that picture, but it looks like you have maybe two wires from the arduino going to the breadboard, and two wires going from the breadboard to that thing on the left. Assuming the thing on the left is the remote control, and that, from what you said shorting its two wires together turns it on, then it's very possible to have, as you say, a transistor that's normally open.

The usual thing would be to use a cheap NPN transistor like a 2n2222 or 2n3904. The collector of the NPN would go to the more positive of the two leads from the remote. The emitter would go to the other remote lead. (Use a DVM to be sure which is which, you can't always count on insulation color!) To ensure that the NPN passes no current when no input is connected, you can put a 50K or 100K resistor across the NPN's emitter and base leads. Then, to get input from the arduino, connect the arduino's ground to the NPN's emitter, and connect the arduino control signal you want to use to the NPN's base through a resistor in the range of 1K to 2K ohms.

A circuit like that should not activate the remote when you disconnect the arduino.