No. As others said, the Tevatron is just the last stage of a chain of accelerators, one that was used (nowadays) just to collide high energy protons and antiprotons and "see what's inside". The neutrinos come from the previous stage (called "Main Injector"): they used to take a few protons off the beam, collide them into a target in a very well defined direction, focus the muons that come from this, get neutrinos from the muon decay and measure them near the detector and in Minnesota, to get an idea of their oscillation (and now, also of their speed). The experiment that does this is called MINOS, and it doesn't depend on the Tevatron at all. Actually, shutting down the Tevatron will help MINOS: they will get more protons, therefore more neutrinos and more data.
By the way, this is exacly the same general arrangement used by the OPERA experiment (the one with FTL neutrinos), where the neutrinos are produced in CERN and measured there and in Gran Sasso.