When a tail loses lift the nose pitches down because the horizontal stabilizer pushes the tail down for stability.
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Actually it is not for stability, but for moment compensation.
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Actually it is for stability....modern jets are still aerodynamically stable..
The Stabilizer is for stability (hence the name), but it does not necessarily have to "push down the tail" for that. Only the momentum arms around the center of gravity, the wing area and the lift over AoA curve slope define whether an aircraft is stable or not, this does not determine the direction of the stabilizer force. To word it very simplistic, an aircraft is stable if the wing loading of the forward wing is greater (taking into account the sign) than that of the aft wing. If the wing loading of the aft wing is negative, this is the trivial case. But it might be also slightly positive (the stabilizer lifts the tail) while still maintaining stability. All modern gliders produce lift on the tail for speeds lower than best glide. However, to compensate for the pitching moment of a cambered airfoil on the main wing, the stabilizer always has to move the nose up (i.e. a Canard has to produce lift, a conventional tail has to push the tail down)
The highest stabilizer downforce required for modern transport aircraft is to compensate the extreme pitching moment of the main wing with full flaps. In cruise, modern aircraft do not produce significant downforce, some even produce slight lift. However, due to complexity trim tanks do disappear again, and we are now moving a bit away again from the idea of the stabilizer producing lift, as the drag penalty of the stabilizer downforce is neglectable in transonic cruise.