How to switch to upright greek in math mode?

You might want to file a feature request to the developer of newtxmath; in the meantime you can change the definitions of the Greek lowercase letters.

\documentclass{report}
\usepackage[T1]{fontenc}
\usepackage[type1]{libertine}
\usepackage[libertine]{newtxmath}
\usepackage{bm} 

\makeatletter
\re@DeclareMathSymbol{\alpha}{\mathord}{lettersA}{11}
\re@DeclareMathSymbol{\beta}{\mathord}{lettersA}{12}
\re@DeclareMathSymbol{\gamma}{\mathord}{lettersA}{13}
\re@DeclareMathSymbol{\delta}{\mathord}{lettersA}{14}
\re@DeclareMathSymbol{\epsilon}{\mathord}{lettersA}{15}
\re@DeclareMathSymbol{\zeta}{\mathord}{lettersA}{16}
\re@DeclareMathSymbol{\eta}{\mathord}{lettersA}{17}
\re@DeclareMathSymbol{\theta}{\mathord}{lettersA}{18}
\re@DeclareMathSymbol{\iota}{\mathord}{lettersA}{19}
\re@DeclareMathSymbol{\kappa}{\mathord}{lettersA}{20}
\re@DeclareMathSymbol{\lambda}{\mathord}{lettersA}{21}
\re@DeclareMathSymbol{\mu}{\mathord}{lettersA}{22}
\re@DeclareMathSymbol{\nu}{\mathord}{lettersA}{23}
\iftx@altnu
  \re@DeclareMathSymbol{\nu}{\mathord}{lettersA}{40}
\fi
\re@DeclareMathSymbol{\xi}{\mathord}{lettersA}{24}
\re@DeclareMathSymbol{\pi}{\mathord}{lettersA}{25}
\re@DeclareMathSymbol{\rho}{\mathord}{lettersA}{26}
\re@DeclareMathSymbol{\sigma}{\mathord}{lettersA}{27}
\re@DeclareMathSymbol{\tau}{\mathord}{lettersA}{28}
\re@DeclareMathSymbol{\upsilon}{\mathord}{lettersA}{29}
\re@DeclareMathSymbol{\phi}{\mathord}{lettersA}{30}
\re@DeclareMathSymbol{\chi}{\mathord}{lettersA}{31}
\re@DeclareMathSymbol{\psi}{\mathord}{lettersA}{32}
\re@DeclareMathSymbol{\omega}{\mathord}{lettersA}{33}
\re@DeclareMathSymbol{\varepsilon}{\mathord}{lettersA}{34}
\re@DeclareMathSymbol{\vartheta}{\mathord}{lettersA}{35}
\re@DeclareMathSymbol{\varpi}{\mathord}{lettersA}{36}
\re@DeclareMathSymbol{\varrho}{\mathord}{lettersA}{37}
\re@DeclareMathSymbol{\varsigma}{\mathord}{lettersA}{38}
\re@DeclareMathSymbol{\varphi}{\mathord}{lettersA}{39}
\makeatother

\begin{document}
What I get: $\theta \bm{\theta}$

What I want: $\uptheta \bm{\uptheta}$ 
\end{document}

Note that \mathbf{\theta} won't do any good, only Latin letters are affected by \mathbf.

---

You can also define \mathup and \mathbfup:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[type1]{libertine}
\usepackage[libertine]{newtxmath}
\usepackage{bm}
\DeclareRobustCommand{\mathup}[1]{\begingroup\changegreek\mathrm{#1}\endgroup}
\DeclareRobustCommand{\mathbfup}[1]{\begingroup\changegreekbf\mathbf{#1}\endgroup}

\makeatletter
\def\changegreek{\@for\next:={%
  alpha,beta,gamma,delta,epsilon,zeta,eta,theta,kappa,lambda,mu,nu,xi,pi,rho,sigma,%
  tau,upsilon,phi,chi,psi,omega,varepsilon,vartheta,varpi,varrho,varsigma,varphi}%
  \do{\expandafter\let\csname\next\expandafter\endcsname\csname\next up\endcsname}}
\def\changegreekbf{\@for\next:={%
  alpha,beta,gamma,delta,epsilon,zeta,eta,theta,kappa,lambda,mu,nu,xi,pi,rho,sigma,%
  tau,upsilon,phi,chi,psi,omega,varepsilon,vartheta,varpi,varrho,varsigma,varphi}%
  \do{\expandafter\def\csname\next\expandafter\endcsname\expandafter{%
    \expandafter\bm\expandafter{\csname\next up\endcsname}}}}
\makeatother

\begin{document}
$\mathup{d}\mathup{\theta}d\theta$

$\mathbfup{d}\mathbfup{\theta}\bm{d}\bm{\theta}$
\end{document}

with xelatex or lualatex you can choose the french style for upright greek letters and have also commands for the bold one:

\documentclass{report}
\usepackage[math-style=french]{unicode-math}
\usepackage{libertine}
\setmathfont{XITS Math}

\begin{document}
What I get: $\theta \mbftheta \alpha \mbfalpha\mbftheta$ \\

What I want: $\theta \alpha\beta \mbfmu$ 
\end{document}