It might be more useful to think of $\Omega_f(M)$ as sitting in a homotopy pullback
$\require{AMScd}$
\begin{CD}
    \Omega_f(M) @>>> M\\
    @V  V V @VVfV\\
   M  @>>\operatorname{Id}> M 
\end{CD} 

Then you can make use of the "Mayer-Vietoris sequence" of homotopy groups
$$\cdots \to \pi_2(M, x_0) \stackrel{\partial}{\to} \pi_1(\Omega_f(M),x_0) \to \pi_1(M, x_0) \times \pi_1(M, x_0) \to \pi_1(M,x_0) \to \cdots$$
described in [the answers here.][1] The last map above is $(a,b)\mapsto a\cdot f_*(b)^{-1}$, which is not a homomorphism in general but is a surjection of pointed sets. This immmediately shows for example that when $\pi_2(M,x_0)=0$ then $\pi_1(\Omega_f(M),x_0)$ is isomorphic to the graph of the homomorphism $f_*:\pi_1(M,x_0)\to \pi_1(M,x_0)$.

To go further you'd have to analyse the connecting map $\partial: \pi_2(M,x_0)\to \pi_1(\Omega_f(M),x_0)$. It may be (though I'm not sure) that the map preceding it is given by $(x,y)\mapsto x-f_*(y)$, in which case $\partial=0$. A good place to read up on this sequence seems to be May and Ponto's *More concise algebraic topology*, Section 2.2 


  [1]: https://mathoverflow.net/q/132339/8103