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.