Use Arduino for switch a 12v powered remote control
A direct connection is not a good idea since there are too many unknowns.  The safest thing is to have the microcontroller activate something that acts like a switch.  A relay definitely does this.

However, an opto-isolator will most likely work too, and will be smaller, cheaper, and take less power.  You should be able to drive it from a microcontroller output with resistor in series.  You don't need speed, so you can use a opto with decent gain, like 3x.  1 mA in yields 3 mA output current capability, which is likely a lot more than the switch has to handle.  You still have to get the polarity right though.  Measure the open circuit voltage across the button you want to simulate pressing.  That will tell you which way around the opto output needs to be hooked up.