Is a three body gravitating system doomed to collapse? Suppose we have two gravitating bodies, which are rotating around each other. They are bodies and are affected by deformation caused by tidal forces. Moving tidal waves suck energy from the axial rotation of these bodies. After some time their axial rotation will be synced with orbital rotation and tidal waves will be stopped. Like Moon orbital rotation around Earth is synced with its axial rotation. Maybe I missed some additional effects but it seems to be a quite accurate assumption.
Except for some special cases, if we have three orbiting bodies, their orbital rotation cannot be synced with axial rotation altogether.  Tidal waves in such a system will exist and suck energy indefinitely. And the only source of energy is potential body energy. Will they fall?
I understand that such effects should be very small; it hardly can be measured in the solar system, because the variation of mass of planets and the Sun causes a much higher effect.
But anyway are there any measures that support the existence of such an effect for something? Or is there theoretical proof that this effect does not work for some reasons I missed?
Since you are talking about real bodies with some extent, yes. The case for point-particles is somewhat different.
Any dissipative system will (by definition) gradually lose energy, and that means that the positions and velocities of the constituent bodies will be confined to (hyper)surfaces in phase space with less and less energy, $E=V(\mathbf{x})+K(\mathbf{v})$. Potential energy $V(\mathbf{x})$ decreases with distance between the bodies and there is a minimum possible separation giving a minimum potential energy, and kinetic energy is always positive $K(\mathbf{v})&gt;0$. So as $E$ declines the distances and velocities have to decline accordingly. Typically things getting close move faster, with kinetic energy increasing as potential energy decreases, but since there is a limit to how much potential energy can decrease before things collide there will eventually be a collision (or, in systems allowing it - not the case here - losing all kinetic energy and ending up in perpetual rest).
(Point particles can cheat in Newtonian gravity by performing weird and physically unrealistic collision or oscillating orbits since they can reduce their potential energy arbitrarily far, see the Painleve conjecture)
In practice dissipation effects are tiny in planetary systems, and for most planets their orbits will instead be disrupted by rare stellar encounters, chaotic resonances with other planets, or mass-loss from the star when it leaves the main sequence. Planets near stars may be dragged in because of tidal effects, but this matters most if one is a few stellar radii away. In the truly long run gravitational radiation will force any planet to spiral in, but the timescales are staggeringly long.
The number of bodies does not change things, but for many-body systems ($N\geq 3$) some bodies can gain enough energy to permanently escape while the rest merge.