Relays are the obvious choice:
If the buttons are a momentary push switch, then I suggest using a relay, and controlling that by energising the coil with an appropriate circuit connected to the Arduino (or similar). Use one relay per button where required. This mitigates the need to know the lattisima's internal circuitry; just replace each switch with a relay (or have them in parallel).
Relays are ideal because any contact bounce of the relay would already be handled by the machine since it would already deal with contact bounce of the switches. Relays are a common addition to Arduinos etc. so many circuits can be found online such as the one below, as well as, no doubt, ready-to-use add-ons.
Otherwise, ideally you will need a circuit diagram for the machine's electronics, so you can determine what circuitry would be compatible with that sensing the button presses. It may be possible to do without a circuit diagram, by either forming one yourself by inspecting the PCB and chips etc; reverse engineering it; or possibly just sensing the voltage at the switch contacts when they are not pressed. Although your mileage with this may vary; I would not recommend it.
To control the Arduino from your phone, perhaps either use a Bluetooth-to-serial module which can be picked up from eBay for a few dollars. Fairly easy to use, just send serial bytes between the module and Arduino. Although this means your phone has to be within 10m (depending on the module).
Another option might be to get an ethernet module for the arduino (I believe they exist) and plug it into your router, and implement something like an http client. My brain is in weekend mode though so this idea needs polishing.
How does all that sound to you?
simulate this circuit – Schematic created using CircuitLab
The above circuit can be modified accordingly to suit what relays you want, whether you use the Arduino or something else, what power rails you have available to you, etc.