The package seems to be allowing the italic correction of the base to leak out, always using a {} at the end of the base seems to fix things
newtx version:
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\documentclass{article}\usepackage[amsthm, nc, varbb]{newtxmath}\usepackage{amsmath}\usepackage{tensor}\begin{document}\[ \tensor{T{}}{^\mu_\nu} \]\[ \tensor{\mathrm{\mathcal{T}}{}}{^\mu_\nu} \]\[ \tensor{\Lambda{}}{^\mu_\nu} \]\[ \tensor{R{}}{^\mu_\nu} \]\end{document}