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. 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
Edit: as Tyrone points out, I wrote down the wrong pullback square. What I should have written down was