  We study the search potential of heavy triplet leptons in Type III Seesaw
mechanism at future high-energy and high-luminosity muon colliders. The impact
of up-to-date neutrino oscillation results is taken into account on neutrino
parameters and the consequent decay patterns of heavy leptons in Type III
Seesaw. The three heavy leptons in the Casas-Ibarra parametrization with
diagonal unity matrix are distinguishable in their decay modes. We consider the
pair production of charged triplet leptons $E^+ E^-$ through both $\mu^+\mu^-$
annihilation and vector boson fusion (VBF) processes. The leading-order
framework of electroweak parton distribution functions are utilized to
calculate the VBF cross sections. The charged triplet leptons can be probed for
masses above $\sim 1$ TeV. The $E^+E^-\to ZZ\ell^+\ell^-$ channel can be
further utilized to fully reconstruct the three triplet leptons and distinguish
neutrino mass patterns. The pair production of heavy neutrinos $N$ and the
associated production $E^\pm N$ are only induced by VBF processes and lead to
lepton-number-violating (LNV) signature. We also study the search potential of
LNV processes at future high-energy muon collider with $\sqrt{s}=30$ TeV.
