I wouldn't call linearity the property you are referring to. The proper name should be pair-wise additivity, i.e., we assume that if the force on body $i$ due to body $j$ alone is ${\bf F}_{ij}$, and that due to body $k$ alone is ${\bf F}_{ik}$, the total force in $i$, due to the simultaneous presence of $j$ and $k$, is $$ {\bf F}_{i} = {\bf F}_{ij}+{\bf F}_{ik}. $$ More in general, for an n-body system, $$ {\bf F}_{i} = \sum_{j=1;j\neq i}^n{\bf F}_{ij}. $$

Pair-wise additivity of the forces is often used in Newtonian mechanics and it was taken for granted by Netwon, but it is not a necessary condition. Indeed it is quite easy to provide examples of more complicated forms of force law. Even more important, although rarely stressed in the textbooks, the most accurate models of the effective forces among atoms or molecules in condensed matter are certainly not pair-wise additive (see the comment at the end).

Probably the most simple example of a force that is not pair-wise additive is the force between two neutral but polarizable particles, say $1$ and $2$. If only these two particles are present the mutual forces are zero: ${\bf F}_{12}=0$ and ${\bf F}_{21}=0$. However, if we introduce a third, charged particle say number $3$, both the original particles get an induced electric dipole and, in addition to the dipole-charge interactions with the charged body, ${\bf F}_{21}\neq 0$ and ${\bf F}_{12} \neq 0$, due to the dipole-dipole interaction.

A couple of final comments are in order:

1. the formal structure of Newtonian mechanics is able to accommodate non-pairwise forces without problems. It is only the expression of the total force on each particle that is more complex. It should be clear that pair-wise non-additivity does not break the second-law relation between total force and acceleration. Simply put, there is nothing like the additivity of the accelerations due to the presence of different external bodies. This has nothing to do with the vector character of accelerations, of course.
2. if the example of the polarizable particles in the presence or absence of a charge seems too artificial, one should remember that the effective interactions among atoms in condensed matter are always originating from an operation of partial trace over electronic degrees of freedom. An example is the well-known Born-Oppenheimer approximation where the interatomic interaction energy contains a many-body term (i.e. non-pairwise interaction) corresponding to the ground state energy of the electrons in the presence of fixed nuclei.

Note: I have edited my original answer to take into account some comments and to add more generality, in particular stressing the superposition principle as a more general principle than the pair-wise additivity, the latter being a particular case of the former.

I wouldn't call linearity the property you are referring to. The proper name superposition principle. In the simplest case, the superposition principle coincides with the pair-wise additivity of the interactions, i.e., we assume that if the force on body $i$ due to body $j$ alone is ${\bf F}_{ij}$, and that due to body $k$ alone is ${\bf F}_{ik}$, the total force in $i$, due to the simultaneous presence of $j$ and $k$, is $$ {\bf F}_{i} = {\bf F}_{ij}+{\bf F}_{ik}. $$ Notice that in the separate case and in the combined case, each pair-wise force (${\bf F}_{ij}$ and ${\bf F}_{ik}$) is a function only of quantities of the corresponding pair. More in general, for an n-body system, $$ {\bf F}_{i} = \sum_{j=1;j\neq i}^n{\bf F}_{ij}. $$ Superposition (or pair-wise additivity) is definitely a mathematical property different from the bi-linearity of the vector sum. The former has to do with the functional dependence of the contributions to the force on the body parameters, the latter with the operations defined on these functions.

Superposition, or the more specific pair-wise additivity of the forces, is often used in Newtonian mechanics and it was taken for granted by Netwon, but it is not a necessary condition. Indeed it is quite easy to provide examples of more complicated forms of force law. Even more important, although rarely stressed in the textbooks, the most accurate models of the effective forces among atoms or molecules in condensed matter are certainly not pair-wise additive (see the comment at the end).

Probably the most simple example of a force that is not pair-wise additive is the force between two neutral but polarizable particles, say $1$ and $2$. If only these two particles are present the mutual forces are zero: ${\bf F}_{12}=0$ and ${\bf F}_{21}=0$. However, if we introduce a third, charged particle say number $3$, both the original particles get an induced electric dipole and, in addition to the dipole-charge interactions with the charged body, ${\bf F}_{21}\neq 0$ and ${\bf F}_{12} \neq 0$, due to the dipole-dipole interaction.

A couple of final comments are in order:

1. the formal structure of Newtonian mechanics is able to accommodate non-pairwise forces without problems. It is only the expression of the total force on each particle that is more complex. It should be clear that pair-wise non-additivity does not break the second-law relation between total force and acceleration. Simply put, there is nothing like the additivity of the accelerations due to the presence of different external bodies. This has nothing to do with the vector character of accelerations and forces, of course.
2. if the example of the polarizable particles in the presence or absence of a charge seems too artificial, one should remember that the effective interactions among atoms in condensed matter are always originating from an operation of partial trace over electronic degrees of freedom. An example is the well-known Born-Oppenheimer approximation where the interatomic interaction energy contains a many-body term (i.e. non-pairwise interaction) corresponding to the ground state energy of the electrons in the presence of fixed nuclei.

I wouldn't call linearity the property you are referring to. The proper name should be pair-wise additive, i.e., we assume that if the force on body $i$ due to body $j$ alone is ${\bf F}_{ij}$, and that due to body $k$ alone is ${\bf F}_{ik}$, the total force in $i$, due to the simultaneous presence of $j$ and $k$, is $$ {\bf F}_{i} = {\bf F}_{ij}+{\bf F}_{ik}. $$ More in general, for an n-body system, $$ {\bf F}_{i} = \sum_{j=1;j\neq i}^n{\bf F}_{ij}. $$

Pair-wise additivity of the forces is often used in Newtonian mechanics and it was taken for granted by Netwon, but it is not a necessary condition. Indeed it is quite easy to provide examples of more complicated forms of force law. Even more important, although rarely stressed in the textbooks, the most accurate models of the effective forces among atoms or molecules in condensed matter are certainly not pair-wise additive (see the comment at the end).

Probably the most simple example of a force that is not pair-wise additive is the force between two neutral but polarizable particles, say $1$ and $2$. If only these two particles are present the mutual forces are zero: ${\bf F}_{12}=0$ and ${\bf F}_{21}=0$. However, if we introduce a third, charged particle say number $3$, both the original particles get an induced electric dipole and, in addition to the dipole-charge interactions with the charged body, ${\bf F}_{21}\neq 0$ and ${\bf F}_{12} \neq 0$, due to the dipole-dipole interaction.

A couple of final comments are in order:

1. the formal structure of Newtonian mechanics is able to accommodate non-pairwise forces without problems. It is only the expression of the total force on each particle that is more complex. It should be clear that pair-wise non-additivity does not break the second-law relation between total force and acceleration. Simply put, there is nothing like the additivity of the accelerations due to the presence of different external bodies. This has nothing to do with the vector character of accelerations, of course.
2. if the example of the polarizable particles in the presence or absence of a charge seems too artificial, one should remember that the effective interactions among atoms in condensed matter are always originating from an operation of partial trace over electronic degrees of freedom. An example is the well-known Born-Oppenheimer approximation where the interatomic interaction energy contains a many-body term (i.e. non-pairwise interaction) corresponding to the ground state energy of the electrons in the presence of fixed nuclei.

I wouldn't call linearity the property you are referring to. The proper name should be pair-wise additivity, i.e., we assume that if the force on body $i$ due to body $j$ alone is ${\bf F}_{ij}$, and that due to body $k$ alone is ${\bf F}_{ik}$, the total force in $i$, due to the simultaneous presence of $j$ and $k$, is $$ {\bf F}_{i} = {\bf F}_{ij}+{\bf F}_{ik}. $$ More in general, for an n-body system, $$ {\bf F}_{i} = \sum_{j=1;j\neq i}^n{\bf F}_{ij}. $$

Pair-wise additivity of the forces is often used in Newtonian mechanics and it was taken for granted by Netwon, but it is not a necessary condition. Indeed it is quite easy to provide examples of more complicated forms of force law. Even more important, although rarely stressed in the textbooks, the most accurate models of the effective forces among atoms or molecules in condensed matter are certainly not pair-wise additive (see the comment at the end).

Probably the most simple example of a force that is not pair-wise additive is the force between two neutral but polarizable particles, say $1$ and $2$. If only these two particles are present the mutual forces are zero: ${\bf F}_{12}=0$ and ${\bf F}_{21}=0$. However, if we introduce a third, charged particle say number $3$, both the original particles get an induced electric dipole and, in addition to the dipole-charge interactions with the charged body, ${\bf F}_{21}\neq 0$ and ${\bf F}_{12} \neq 0$, due to the dipole-dipole interaction.

A couple of final comments are in order:

1. the formal structure of Newtonian mechanics is able to accommodate non-pairwise forces without problems. It is only the expression of the total force on each particle that is more complex. It should be clear that pair-wise non-additivity does not break the second-law relation between total force and acceleration. Simply put, there is nothing like the additivity of the accelerations due to the presence of different external bodies. This has nothing to do with the vector character of accelerations, of course.
2. if the example of the polarizable particles in the presence or absence of a charge seems too artificial, one should remember that the effective interactions among atoms in condensed matter are always originating from an operation of partial trace over electronic degrees of freedom. An example is the well-known Born-Oppenheimer approximation where the interatomic interaction energy contains a many-body term (i.e. non-pairwise interaction) corresponding to the ground state energy of the electrons in the presence of fixed nuclei.