Hacking IKEA table - OR gate needed
The simplest option is to use an N-Ch MOSFET to simulate a push button operation. 

With the source connected to GND, the drain connected to the button, then the gate becomes an electrically controlled push button. This is a classic open-drain output circuit. As a circuit it should work to interface with most circuits that use a push-button-to-ground approach.

You can drive the MOSFET gate with anything, like your MCU output. A pull-down resistor (from gate to source) would be a good addition to prevent the MOSFET turning on when the MCU pins are high impedance.

The reason for this suggestion is two-fold:


Differences in IO voltages between your microcontroller and the other circuit will not have any impact as the MCU pin is not directly driving the circuit.
You can still push the existing button to control the table without risking damage if the MCU output was ever set high.