Could there be a "massive gravity" theory? If we talk about a "quantum theory" of General Relativity, we know that the particle that mediates the gravitational force would be the so called Graviton, a massless particle with spin-$2$.
I wonder if (and if it would work) there could be also a different theory like a "massive gravity", whose force is mediated by a massive gauge boson (maybe not that massive too, even a small mass). What would it implicate?
 A: The reason why the graviton is proposed to be massless is that the range the gravitational force acts on is infinite and falls as 1/r. The full derivation of this requires a lot of knowledge of relativistic quantum field theory but I will try to motivate this using basic quantum mechanics and special relativity principles. 
The Heisengberg Uncertainty Principle tell us that 
$$ \Delta E \Delta T ≥ \hbar $$
This then can be interpreted as follows: we can borrow energy $\Delta E$ from the universe as long as we give it back within time $\Delta T$. 
From that follows 
$$\Delta T = \frac{\hbar}{mc^2}$$
where m is the mass of the particle that mediates the interaction, and c is the speed of light, the maximum speed this particle moves at. (We used $E = mc^2$ here)
The range of the force is then proportional to $\Delta T$ because it is the distance that our mediating particle can travel in this time. 
Now we know that gravity acts on an infinite range, from that follows that $\Delta T = \infty $ and therefore, m = 0.
So, until we find something contradictory with that gravity acts on an infinite range, this forces us to choose massless gravitons when we talk about a gravity-mediating particle.
A: It is well understood that if gravity is a 'long range force' its quanta's mass is zero. That is the graviton. Excuse the term in quotation marks, its the easiest way to say what is known. It really means the gravitational field, at infinity in aan asymptotic flat spacetime goes like 1/r
But the fact is that the graviton mass could indeed be greater than 0, but very very small, and all the observations and measurements ever done on gravity would not detect any deviation. The current limits on its mass from all the observations/measurements are that the mass must be smaller than 10 to the minus 22 ev. That's is extremely small. The most stringent limits have been set by astronomical observations. But we are soon to to be observing possible effects, and will be able to set the limit lower by orders of magnitude, if indeed it is 0. Those upcoming measurements will be from observations of gravitational waves in the space-based eLISA satellites that will constitute a 3 leg interferometer in space with interferometer led aisles of 1 million kilometers. It is to be launched in the next very few years. That greater spacing will increase the sensitivity and allow for much longer wavelengths to observe gravitational waves. It'll be a significant sensitivity improvement over the ground based 5 Kms leg sizes that did the first direct detection of gravitational waves in 2016, announced in February of this year. 
The new limits will be based on looking for dispersion in observed gravitational waves - meaning slightly different velocities measured at different frequencies. It will be that sensitive. It'll also observe those waves from dual merging supermassive black holes much longer, up to months, so a lot of data to do statistical averaging over also. 
Those measurements will be looking for plenty other possible deviations from general relativity, and it will do it in the realm of strong gravity where first post Newtonian approximations will not do. It'll also be able to detect higher multipole moments from those black holes in the merger phase and ringdown, and thus see both dynamic effects of the settling to the no hair Kerr black holes. It'll see any deviations from the no hair theorem, up to its sensitivity limit. It'll be able to check other gravity theories such as strong gravity. Some scalar massive spin 0 plus tHe spin 2 theories will also be able to be ruled out, or evidence found for that possibility. It'll be able to see cosmological gravitational waves, and any waves emitted by some Big Bang relics such as cosmic string, if any existed. 
Over time there will be even bigger gravitational wave space based 'observatories' with longer legs and even better sensitivity
The point is that yes there are some theories that allow non zero mass gravitons and which have not been ruled out, and those will be explored, and if the mass is truly 0 the mass limits will be made more stringent. At the very least we will be observing a lot of the details of those waves front many astrophysical and cosmologically predicted or expected objects. Someone named some of those measurements as 'gravitational spectroscopy' (sorry I can remember who but it was in relation to the quasi-normal modes in black hole ringdown after merger. 
Just like the neutrino is now believed to have a very small mass, we still don't know enough about gravity, nor quantum gravity, to know for sure, to full accuracy and theoretical consistency, for the graviton. Still, for now, all measurements and observations, and accepted theory, have found it to be a big zero. 
Just tune in over time. 
A: Yes. A gravitational field theory mediated by a massive spin-2 gauge boson (massive graviton) is called massive gravity. Actually, this is a very active field in modified gravity and theoretical physics. In the following, I will review some features as well as some of the most important results of this theory.

*

*In such a theory, according to the field theory, a massive graviton has five-degrees of polarization. If you consider an explicit tiny mass term for gravitons, according to the different models of massive gravity, you always arrive at a Yukawa-like potential for a point source as
$$U(r) \propto \frac{-1}{{4\pi r}}{e^{ - m_gr}},$$
where $m_g$ is the mass of graviton. By treating massive gravity as a quantum effective field theory, it can also be confirmed that and the force is always attractive. Furthermore, assuming an extremely tiny mass for graviton ($m_g \ll 1$), the effect of the exponent ($e^{-m_gr}$) in the potential energy could be even undetectable in solar scales (as well as in larger scales) if one demands that $m_g {r} \sim 0 $.


*The first attempt for formulating such a theory was made by Fierz and Pauli (1939), known as Fierz-Pauli (FP) theory. The FP theory is sometimes referred to as linearized massive gravity. In this theory, using the weak-field approximation of gravity (by linearizing the theory around the flat background spacetime), i.e.,
$${g_{\mu \nu }} = {\eta _{\mu \nu }} + {h_{\mu \nu }},\,{\rm{where}}\,\,\,\left| {{h_{\mu \nu }}} \right| \ll 1,$$
the action describing a free massive graviton may be written as
$${\cal S}_{{\rm{FP}}}^{(L)} = \int {{d^4}x\left( { - \frac{1}{2}G_{(L)}^{\mu \nu }{h_{\mu \nu }} - \frac{1}{2}m_g^2\left( {{h^{\mu \nu }}{h_{\mu \nu }} -  h^2 } \right)} \right)},$$
where $h=h_\mu ^\mu$, $\kappa =8 \pi G/c^4$ and $G_{(L)}^{\mu \nu }$ is the linearized Einstein tensor. The FP field equation is obtained as
$${({\partial ^2} - {m_g^2}){h_{\mu \nu }} = 0},$$
which reminds us of field equations for massive scalar and massive vector fields, respectively, ${({\partial ^2} - {m^2}){\phi}= 0}$ and ${({\partial ^2} - {m^2}){A_\mu}= 0}$. But, since this theory in the massless limit does not reduce to the linearized general relativity (which, e.g., predicts gravitational waves or bending of light), it was gradually discarded as an inconsistent theory, but still theoretically attractive.


*Finding a consistent theory of massive gravitational field theory free from pathological behaviors was one of the long-standing, important problems in theoretical physics for decades from the pioneering work of Fierz and Pauli. In the last decade, eventually, a self-consistent, ghost-free theory of massive gravity has been discovered by de Rham, Gabadadze, Tolley (2010-2011). This theory is widely known as (dRGT) massive gravity and its action is given by
$${{{\cal S}_{{\rm{MG}}}} = \int {{d^4}x\sqrt { - g} } \left[ {R - \frac{1}{4}m_g^2\sum\limits_{i=1}^{} {{c_i}{{\cal U}_i}(g,f)} } \right]},$$
where the term "$-\frac{1}{4}m_g^2\sum\limits_{i=1}^{}{{c_i}{{\cal U}_i}(g,f)}$" is a mass term (sometimes is called interaction potential). This term is essential for building an appropriate mass term for spin-2 gauge bosons, which is defined based on a (non-physical) reference metric $f_{\mu \nu}$. Note that any massive gravity theory inevitably include an auxiliary reference metric. The resultant gravitational field equations are found to be as
$${G_{\mu \nu }} + m_g^2{{\cal X}_{\mu \nu }} = 0,$$
where the term "$m_g^2{{\cal X}_{\mu \nu }}$" represents the mass of graviton.


*The mass of graviton must be tuned to extremely small values for the theory of massive gravity to be viable. In fact, we already have some valuable data that can restrict the mass of graviton. Observations of LIGO have constrained a lower bound on the graviton Compton wavelength as
$${{\lambda _{{\rm{graviton}}}} > 1.6 \times {{10}^{13}}{\rm{km}}}.$$
Assuming that gravitons are dispersed in vacuum like massive particles, i.e.,
$${\lambda _{\rm{graviton}}} = \frac{h}{{{m_g}\,c}},$$
this means an upper bound exists for graviton's mass as
$${{m_{{\rm{graviton}}}} \le {{10}^{ - 22}}eV/{c^2} \sim {{10}^{ - 38}}g}.$$ This is extremely small, well below the limit of being detectable. So, the effective potential energy in massive gravity, $U(r) \propto \frac{-1}{{4\pi r}}{e^{ - m_gr}}$, is nicely justified.


*In massive gravity, the speed of gravitational waves will be slower than the speed of light, though it is very close to the speed of light. On the other hand, massive gravity theories are generally non-renormalizable just as for general relativity. This is because any massive gravity theory only modifies general relativity at large scales and should be regarded as an effective field theory.


*The dRGT massive gravity has been formulated with respect to a flat, non-dynamical reference metric. It is possible to give dynamics to the reference metric and the resulting theory is called as bimetric gravity (also known as bigravity), which is also free from pathological behaviors.


*A number of observations support that the theory of (bi)massive gravity is phenomenologically viable. For example, in this paper (and also this one) the authors have shown that all current observations related to dark matter can be explained by the presence of a heavy spin-2 particle (the authors claimed that this theory automatically contains a perfect dark matter candidate). As another example, in this paper the authors has shown that the (bimetric) massive gravity is a strong candidate to explain the accelerated expansion of the Universe.
Finally, to complete the discussion, I've summarized some of the important differences between general relativity and massive gravity theories, in the table below.




Properties
General relativity
Massive gravity




Graviton's degrees of polarization
2
5


Speed of gravitational waves
$v=c$
$v<c$


Ghost free
yes
in some cases (e.g., dRGT and bigravity models)


Gravitational potential energy (in 4-dimensions)
$U(r) \propto -\frac{1}{r}$
$U(r) \propto -\frac{1}{r} e^{-m_g r}$


Renormalizability
no
generally no (But, topologically massive gravity in 3-dimensions is renormalizable)


Diffeomorphism symmetry
yes
can be restored by Stückelberg fields


Dark energy component for explaining the accelerated expansion of the Universe
needed
not needed




Other References
[1] C. de Rham, Massive gravity, Living Rev. Relativity 17 (2014) 7
[2] A. Zee, Quantum field theory in a nutshell, Princeton university press (2010).
[3] K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671
[4] M.D. Schwartz, Quantum field theory and the standard model, Cambridge University Press (2014)
A: Observation puts an upper limit to graviton mass of 10$^{-23}$ eV/c$^2$. https://physicsworld.com/a/motions-of-the-planets-put-new-limit-on-graviton-mass/
