Switch – How to remove old light switches, leave circuit always on

You say that, when you release the button, the contact opens again (resistance goes to infinity). But the light stays on, right? That means there must be some device which has a "memory" of the last action: you push a button, the light goes on and stays on, until you press the button again (or another button). I'm sure there is a device in the cabinet (Dutch: "schakelkast") which toggles its output on every pulse it receives, and which in this way controls the lamp.
The wire on the top right of the switch also hints in this direction; it's a signal wire, it's too thin to be a power wire. My hunch is that it carries the pulses to the actual switching device (I don't know what it's called).

This explanation assumes you are trying to control both sets of lights at the same time and not independently.

If they are on the same circuit, having separate hot wires to each switch is really redundant. The return wires from the switches are not.

There is no problem ganging both lighting fixtures on the same switch so long as the switch is rated for the power the fixtures draw. Most modern dimmers are rated 150, 300, 600 or 1000 watts. You need to make sure that the two fixtures, added together, are somewhat lower than the switch rating. If not, your gonna need a bigger switch!

All you need to do is hook one of the hot lines to the switch and cap the other hot line with a wirenut. Then pigtail both of the two switched lines (that return to the fixtures) to the switch output side. Do be sure to attach ground, for safety and to reduce the chance of random hum.

All of this assumes that the dimmers are suitable to the type of lighting (special dimmers are needed for many CFL and LED bulbs), that they are single pole (not three way), and there is no special feature requiring a neutral.

SUPPLEMENT

As Tester101 needs to regularly remind me, current code calls for a neutral wire at all switches. You have an extra wire in the switch box (the redundant hot) that could be converted to a neutral if you ever need it. You would need to switch it over at the fixture to the neutral white line, mark it at the fixture to show it is neutral (e.g., with white tape) and mark it at the switch box as neutral (also white tape).