The collisions produce a shower of decay chain particles that impinge on the calorimeters (sensors that detect particle energy) surrounding the experiments. Particles in the beam that are not scattered every which-way by collisions can be diverted to a beam dump. Beam dumps are large targets, several meters long, that are designed to absorb the astonishing kinetic energy of the beam.
I recall during the CMS (Compact Muon Solenoid, one of the LHC experiments) talk broadcast today that the CMS calorimeter pixels have been noticeably degrading over time due to the high energy particles hitting them. This is probably because the incident particles are of sufficient energy to introduce crystallographic defects into the lead tungstate crystals used in the calorimeters.