I am following Lee's Introduction to Smooth Manifolds, in which he considers $n$ particles with masses $m_1, \ldots, m_n$ and where the position of the $i$th particle is given by $q^i_1, q^i_2, q^i_3$. The time evolution of this system can be described as a curve in $\mathbb{R}^{3n}$: $$q(t) = \big(q_2^1(t), q_2^1(t), q_2^1(t), \ldots, q_2^n(t), q_2^n(t), q_2^n(t)\big).$$ Letting $M = (M_{ij})$ denote the $3n \times 3n$ diagonal matrix whose diagonal entries are $$(m_1, m_1, m_1, m_2, m_2, m_2, \ldots, m_3, m_3, m_3)$$ and $$F(q) = \big(F_1(q), \ldots, F_{3n}(q)\big)$$ the $3n$ components of the forces. Then from Newton's second law we have that $$M_{ij}\ddot{q}^j(t) = F_i\big(q(t)\big)$$ where the summation convention has been applied. Lee remarks
We assume that the forces depend smoothly on $q$, so we can interpret $F(q) = \big(F_1(q), \ldots, F_{3n}(q)\big)$ as the components of a smooth covector field on $Q$.
How is $F(q)$ a covector field? It seems we never uses tangent vectors above nor does $F$ act on them.
Trying to expand the discussion so far, in particular @Deane and @user10354138 comment on Hamilton/Newton mechanics. Forces would modify momentum, affecting $p$. I would recommend sections 3.11 and 3.12 of Marsden and West but I believe their equation (3.1.3) has an obvious typo, mixing $p$ and $q$, if you compare with equation 1.4.3 on page 377 of that article, and usual convention for Hamilton vector fields, for example wikpedia. The equations should read $$\dot{q} = X_{q}(q, p) = \frac{\partial{H}}{\partial p},\\ \dot{p} = X_{p}(q, p) = -\frac{\partial{H}}{\partial q} + f_H(q, p). $$ In this system, if you follow the flow then the energy $H$ is not conserved but changed by $$\frac{d}{dt}H(q(t), p(t)) = \frac{\partial H}{\partial q}\dot{q} + \frac{\partial H}{\partial p}\dot{p} = \frac{\partial H}{\partial q}(\frac{\partial{H}}{\partial p}) + \frac{\partial H}{\partial p}(-\frac{\partial{H}}{\partial q} + f_H(q, p)) = \frac{\partial H}{\partial p}.f_H(q, p). $$ Here, $\frac{\partial H}{\partial p}$ is tangent-like, so $f_H$ should be cotangent-like for the contraction gives us a scalar.
Also, $f_H(q, p)$ is an element of $T^*_qQ$ for the same reason for the affinity of connections. In general, if $E$ is a vector bundle over $Q$, if $(\Delta_q, \Delta_e)\in T_{q, e}E$ is a tangent vector at $(q, e)\in E$ and $e_1\in E$ projects to $q$ then $(\Delta_q, \Delta_e + e_1)$ is another tangent vector at $(q, e)\in E$. Take $E = T^*Q$, $e_1=f_H(q, p)$ in this case.
To answer what does $f_H$ acts on to be a covector, it pairs with $\dot{q}=\frac{\partial H}{\partial p}$ to be the rate of change of energy, I think it is called power.