Should subscripts in math mode be upright?

Using "all italics" is unfortunately an often committed sin. You should italicize only variables.

Everything else should be upright. For example:

• function names (sin, cos, log, ln etc...)
• dimensionless numbers (Re, Pr, Ra...)
• exact infinitesimal increments (dx, dy et... in BOTH integrals and differentials)
• descriptive text
• all descriptive variable indices (unless they are also variables)

Exceptions to this rule may still apply, i.e. the Euler-number $e$ is no variable, but still traditionally written in italics.

Some of the above was stolen from the "Please Make A Note" blog. More detailed information can be found in "Typefaces" from NIST.

\text is not the right command to use, as its argument will be typeset in the current font, which may be italics.

\usepackage{amsmath} % for extended version of \textup
\newcommand*{\Boltzmann}{k_{\textup{B}}}

$\vec{F}_{\textup{in}}+\vec{F}_{\textup{out}}=\vec{0}$

What subscripts are to be set in upright font and what in italics is well explained by Boffin. (I disagree about the d for the differential, though, but that's another story: be consistent, that's all for this particular aspect.)

For me it depends on the context. Using your example, I would agree that you should use $k_{\text{B}}$ since it is short for $k_{\text{Boltzmann}}$. Note the use of \text{...} from the amsmath package, that does a decent job of sticking to the math mode font size, depending on the context.

On that note, $k_{Boltzmann}$ "looks strange" because (La)TeX typesets it like it does regular math, which in this case is the variable B, followed by the variable o, followed by the variable l, and so on. My use of the term "variable" may be incorrect. The point I'm trying to make is that each letter is treated on it's own in math mode, rather than combining them like is done in text mode. The spacing is correct as-is in math mode, it just doesn't look like text.

With this discussion in mind, I hope you agree that $x_i$ should remain $x_i$. Of course, if you are referring to i in your text somewhere (instead of $i$), then it should be upright. IMHO, it is all (mostly?) driven by context.