Long-range interactions arise naturally in string theory by the exchange of open and closed strings that are in the massless state. For long distances, the correlation produced by massive strings will drop out exponentially with typical length $L\gg\hbar c/m_s\sim l_s$, where $l_s$ is the string length and $m_s$ is the lowest massive state. So, for large distances, just the massless string state will be important, and for bosonic closed string these states will be effectively described by the following action:
$$
S=\frac{1}{2\kappa_0}\int d^Dx(-G)^{1/2}e^{-2\Phi}\{-\frac{2(D-26)}{2\alpha'}+R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_{\mu}\Phi\partial^{\mu}\Phi+\mathcal{O}(\alpha')\}
$$
Where $\Phi$ is the dilaton, $R$ is the Ricci scalar and $H_{\mu\nu\lambda}$ is the field strength of the $B$-form field. This theory describes interactions that does fall as $r^{1-d}$, where $d$ is the number of non-compact dimensions.
For bosonic open strings, with Chan-Paton degrees of freedom, we have Yang-Mills theory for large distance:
$$
S=\frac{1}{g_o'^2}\int d^{26}x\{-\frac{1}{2}Tr(D_{\mu}\varphi D^{\mu}\varphi)+\frac{1}{2\alpha'}Tr(\varphi^2)+\frac{2^{1/2}}{3\alpha'^{1/2}}Tr(\varphi^3)-\frac{1}{4}Tr(F_{\mu\nu} F^{\mu\nu})\}
$$
where $\varphi$ is the tachyon, $F_{\mu\nu}$ is the strength tensor of the gauge $A_{\mu}$.
There are many more sources of such long-range interactions, and they are all described by the massless states of a given string theory. This is so because the exchange of massless objects produces, at tree-level, S-matrices of the type:
$$
\langle p',k \rvert S \lvert p,k \rangle \rvert_{conn} = -\mathrm{i}\frac{e^2}{\lvert \vec p -\vec p'\rvert^2 - \mathrm{i}\epsilon}(2m)^2\delta(E_{p,k} - E_{p',k})(2\pi)^4\delta(\vec p - \vec p')
$$
with is equivalent to a Coulomb potential. See this for more information.
Actually, in the case of superstrings, the long-range features of the theory are used to distinguish each of the five consistent superstring theories:
and all these long-range theories do have the interactions you are wondering about.
In the case of D-branes, we can have massless states trapped on a stack of D-branes, leading to gauge theories inside. See this and this for more information.
Now, if you are wondering if we can describe this interaction by doing a full stringy calculation, i.e. calculating the scattering amplitude via perturbative string theory, the answer is obviously yes. The only thing that changes is that there are annoying massive terms that will show up in each scattering amplitude, terms that do not contribute to long-distance (low energy), and terms proportional to $\alpha'$.
Example: the scattering amplitude at tree level of three gauge bosons from Chan-Paton is given by
$$
\mathcal{A}(k_1,a_1,e_1;k_2,a_2,e_2;k_3a_3,e_3)=ig_o'(2\pi)(2\pi)^{26}\delta^{26}(\sum_{i} k_i)((e_1\cdot k_{23}) (e_2\cdot e_{3})+\\+(e_2\cdot k_{31}) (e_1\cdot e_{3})+(e_3\cdot k_{12}) (e_2\cdot e_{1})+\frac{\alpha'}{2}(e_1\cdot k_{23})(e_2\cdot k_{31})(e_3\cdot k_{12}))
$$
the last term does not contribute to long distance.
If you want to know about off-shell amplitudes, there is String Field Theory.
