How to replace push buttons on an intercom?
I have not worked on this particular system before, but I have worked on some older intercom systems. The current going through these buttons won't be much. To be sure, get ones that are fine to handle a few hundred mA and you will be fine. 

It looks like it is a 16V system, but again, to be safe, get a  24V rated button. 

The next bit is where you will have to be careful. You can see where the buttons are connected, but your new buttons are likely to be physically longer than the current ones. You will need to desolder the current buttons, and remove them. If you can, move the PCB out of the way. 

Next, you will have to make a new hole for the new buttons to fit in. Chances are your new buttons will be longer than the current ones. Make sure you make a hole that the button can sit in, and be securely fixed with the nut on the back. Then simply solder wires from the button to the PCB. Use your photographs to make sure each one is wired to the correct place.

This could well be more trouble than it is worth and there is a chance you could damage the system, and break it. Only attempt this if you are 100% confident you can do it, or if you have the money to pay for a callout fee for it to be repaired if something goes wrong.

Going by the photos, you can make out the manufacturer; LeeDan. The model number is also handily written; IR081-C.

Googling this will take you to THIS PAGE where the company does actually offer their own 'facelift' upgrades to this panel. If you are intent on doing a DIY job, then maybe THIS DATASHEET may be of use.