Force, in its Newtonian sense, is an integral part of general relativity. General relativity attributes the effects of gravitation and inertia to a dynamic 3D metric and a "dynamic" time lapse. But all other forces remain as they were. In fact we can say that it's force, momentum, and energy – not mass – that cause curvature and are affected by curvature.
This is clear from the Einstein equations:
$$
\pmb{G} =
\kappa\pmb{T} \equiv
\kappa\begin{pmatrix}
\text{energy density}&\text{energy flux}\\
\text{momentum density}&\text{stress}
\end{pmatrix}
$$
where stress is basically the classical force/area and includes all standard contact forces such as pressure, friction, tension, viscosity, and so on.
This is even clearer if we write the Einstein equations in 3+1 form, choosing a particular but arbitrary frame in spacetime. Such a frame is chosen in two steps. First we choose a slicing of spacetime into 3D spacelike hypersurfaces, which can be labelled by an arbitrary parameter $t$. Then we (arbitrarily) identify points among such hypersurfaces – we are basically saying "I decree that this point on the hypersurface $t'$ is the same as this point on the hypersurface $t''$", thus arbitrarily decreeing what is at rest and what isn't. Such a choice is represented by a so-called lapse function $\lambda$, which relates proper time with our arbitrary labelling $t$, and a so-called shift vector $\pmb{v}$, which connects points "at rest" across the 3D spacelike hypersurfaces. For details and much better explanations see for example York (1979), Smarr & York (1978), Smarr & al. (1980), Marsden & Hughes (1994) § 2.4, Misner & al (1973). Note that this notion of frame is much freer than in Newtonian mechanics, where frames are traditionally chosen with respect to universal time and an unchangeable spatial metric.
The Einstein equations in 3+1 form can be found for example in Gourgoulhon (2012) chs 4–6, Alcubierre (2008) ch. 2, Rezzolla & Zanotti (2013) ch. 7, or Wilson & Mathews (2007) ch. 1:
$$
\begin{align}
\partial_t \pmb{g} &= -2 \pmb{K} \pmb{g}
\tag{1}\label{1}
\\
\partial_t \pmb{K} &= \pmb{K}\ \mathrm{tr}\pmb{K}
- \pmb{K}^2 +
\pmb{R}
+
\frac{\kappa}{2}\ (\mathrm{tr}\pmb{\tau} - \epsilon) - \kappa\ \pmb{\tau}
\tag{2}\label{2}
\\
\partial_t \epsilon &= \epsilon\ \mathrm{tr}\pmb{K} - \nabla\cdot \pmb{p}
+ \mathrm{tr}(\pmb{K}\pmb{\tau})
\tag{3}\label{3}
\\
\partial_t\pmb{p} &= \pmb{p}\ \mathrm{tr}\pmb{K}
- \nabla\cdot\pmb{\tau}
\tag{4}\label{4}
\end{align}
$$
In these equations we have chosen a free-falling frame of reference (unit lapse function $\lambda$ and zero shift vector $\pmb{v}$).
$\partial_t$ (really a Lie derivative) is the time derivative with respect to a place fixed in this frame;
$\pmb{g}$ is the 3-metric of a 3D spacelike slice, with Ricci curvature $\pmb{R}$ and covariant 3-derivative $\nabla$;
$\pmb{K}$ is the extrinsic curvature of such a slice within the 4D spacetime (note that it is independent of $\pmb{g}$);
$\pmb{p}$ is 3-momentum density, which is equivalent to an energy flux in general relativity;
$\epsilon$ is energy density;
and finally
- $\pmb{\tau}$ is the ordinary stress, that is, any kind of contact force per area.
Equation $\eqref{2}$ shows how 3D space changes its curvature with time: this change depends on the distribution of energy density and of contact forces, $\pmb{\tau}$, within the slice. Equation $\eqref{4}$ is basically the classical Newtonian equation relating the change of momentum to force; one can show that the term $\pmb{p}\ \mathrm{tr}\pmb{K}$ corresponds to convection of momentum, also present in Newtonian continuum mechanics (see eg Marsden & Hughes 1994: $\pmb{K}$ is related to the symmetrized gradient of the velocity $\pmb{u}$ of a matter element, $\tfrac{1}{2}(\nabla\pmb{u} +\nabla\pmb{u}^{\intercal})$). "Fictitious forces" such as gravitation or Coriolis appear in this equation when we choose a different frame, represented by a non-unit lapse function $\lambda$ and a non-zero shift vector $\pmb{v}$. The general equation in an arbitrary frame is (warning, I may have some signs wrong)
$$
(\partial_t - \mathrm{L}_{\pmb{v}})\pmb{p} =
\lambda\ \pmb{p}\ \mathrm{tr}\pmb{K}
- \nabla\cdot(\lambda \pmb{\tau}) - \epsilon\ \nabla\lambda
$$
where $\mathrm{L}$ is the Lie derivative. See for example eqn (6.20) in Gourgoulhon (2012). The term $\epsilon\ \nabla\lambda$ is related to the gravitational force in the Newtonian form $\rho\ \nabla\Phi$, where $\rho$ is mass density. Gourgoulhon (2012), ch. 6, again gives a nice dicussion of the Newtonian limit of the equations above.
So general relativity says that the relation between force and rate of change of momentum is affected by curvature (the term with $\mathrm{tr}\pmb{K}$) and by our arbitrary choice of reference frame (the terms with $\lambda$ and $\pmb{v}$). Our reference frame actually affects all equations above (see cited references).
Regarding geodesic motion: most matter actually does not follow geodesic motion. The parts of our bodies, the parts of the computer or phone in front of us, water in a pipe, the oceans, the plasma in stars, do not follow geodesic trajectories. This is clear from the equations above (to which we must add an equation of conservation of matter). In fact, geodesic motion is quite a tricky concept in general relativity. We can't consider pointlike particles – "Dirac deltas of rest mass", so to speak – because they would generate singularities: general relativity is at heart a field or continuum theory, not a particle theory. Matter must always be considered as "smeared out". If some smeared-out matter can be completely surrounded by a world-tube with small cross section, then such a world-tube is approximately described by a geodesic. But small matter elements within such world-tube are typically not following geodesics. On this problem see for example the review by Infeld & Schild (1949) and Geroch & Soo Jang (1975).
The role of force in general relativity is often a confusing topic. In my opinion part of the problem is that most physics curricula don't teach Newtonian continuum mechanics anymore – yet general relativity is based on it, as clear from the important role that classical stress $\pmb{\tau}$ and its 4D generalization, the stress-energy-momentum tensor $\pmb{T}$, play in it. I recommend the text by Marsden & Hughes (1994), which gives an overview of Newtonian continuum mechanics with many connections with Lorentzian relativity, and also the beautiful article by Eckart (1940). The brilliant text by Frankel (1979) also gives a deeper understanding of the transition "from force to curvature".
References
Alcubierre (2008): Introduction to 3+1 Numerical Relativity (Oxford).
Eckart (1940): The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid. Phys. Rev. 58, 919.
Frankel (1979): Gravitational Curvature: An Introduction to Einstein's Theory (Freeman).
Geroch, Soo Jang (1975): Motion of a body in general relativity. J. Math. Phys. 16, 65.
Gourgoulhon (2003): 3+1 Formalism in General Relativity: Bases of Numerical Relativity (Springer).
Infeld, Schild (1949): On the motion of test particles in general relativity. Rev. Mod. Phys. 21, 408.
Marsden, Hughes (1994): Mathematical Foundations of Elasticity (Dover).
Misner, Thorne, Wheeler (1973): Gravitation (Freeman).
Rezzolla, Zanotti (2013): Relativistic Hydrodynamics (Oxford).
Smarr, Taubes, Wilson (1980): General Relativistic Hydrodynamics: The Comoving, Eulerian, and Velocity Potential Formalisms. In Tipler (ed.): Essays in General Relativity: A Festschrift for Abraham Taub (Academic Press), 157.
Smarr, York (1978): Kinematical conditions in the construction of spacetime. Phys. Rev. D 17, 2529.
Wilson, Mathews (2007): Relativistic Numerical Hydrodynamics (Cambridge).
York (1979): Kinematics and dynamics of general relativity. In Smarr (ed.): Sources of Gravitational Radiation (Cambridge), 83.
