Feeding latex or pdflatex with
\documentclass{article} \usepackage{newtxmath} \begin{document} \(\mathit{\vec{var}}\) \end{document} results in
var
without any arrow and without any warning on the console. (One has to look into the log to see “Missing character: There is no ® in font cmti10!” and wonder what ® means 🙂.) Of course, the problem can be circumvented by dropping \usepackage{newtxmath} or exchanging the order of calling \vec and \mathit. However, since the calls to the two macros may be hidden inside other macros (and for my larger, non-minimal LaTeX sources, they are indeed hidden inside other macros), it would be better to solve the problem already on the level of the preamble or in the NewTX package or by patching it in such a way that the user's main document needn't be changed. Any idea how to do this? I informed the NewTX maintainer. (However, he didn't reply within two months, so whoever wishes to ping him, please feel free…)
PS. For xelatex/lualatex and TeX Gyre Termes (Math), cf. \mathit{\vec{…}} silently swallows up the vector when using xelatex/lualatex, unicode-math, and TeX Gyre Termes (Math) .
PPS. The best workaround so far is given by Vincent's answer http://tex.stackexchange.com/a/643761 .

\mathit{\vec{var}}? It should be\vec{\mathit{var}}.\mathit{\vec{foo}}says that the vectorfoois to be typeset as a term variable (in the context of some formula).\vec{\mathit{foo}}would mean: typeset the variablefooas a vector or typeset the application of the overrightarrow operation (which is defined in the book I typeset) to the variablefoo. The semantic meaning is different. In fact, semantically, I have both: (1) typeset a vector as a term variable (rather than, e.g., as a term constant, or as an n-ary operation, or as a sort name) and (2) apply the overrightarrow operation (whatever it means) to a term variable.