  Dynamics of disentanglement as measured by the tripartite negativity and Bell
nonlocality as measured by the extent of violation of the multipartite
Bell-type inequalities are investigated in this work. It is shown definitively
that for the initial three-qubit Greenberger-Horne-Zeilinger (GHZ) or {\it W}
class state preparation the Bell nonlocality suffers sudden death under the
influence of thermal reservoirs. Moreover, all the Bell-nonlocal states are
useful for nonclassical teleportation, while there are entangled states that do
not violate any Bell-type inequalities, but still yield nonclassical
teleportation fidelity.
