The possibility that the so-called lithium problem, i.e. the disagreement between the theoretical abundance predicted for primordial $^7$Li assuming standard nucleosynthesis and the value inferred from astrophysical measurements, can be solved through a non-thermal BBN mechanism has been investigated by several authors. In particular, it has been shown that the decay of a MeV-mass particle, like, e.g., a sterile neutrino, decaying after BBN not only solves the lithium problem, but also satisfies cosmological and laboratory bounds, making such a scenario worth to be investigated in further detail. In this paper, we constrain the parameters of the model with the combination of current data, including Planck 2015 measurements of temperature and polarization anisotropies of the CMB, FIRAS limits on spectral distortions, astrophysical measurements of primordial abundances and laboratory constraints. We find that a sterile neutrino with mass $M_S=4.35_{-0.17}^{+0.13},MeV$ (at $95%$ c.l.), a decay time $tau_S=1.8_{-1.3}^{+2.5}cdot 10^5,s$ (at $95%$ c.l.) and an initial density $bar{n}_S/bar{n}_{cmb}=1.7_{-0.6}^{+3.5}cdot 10^{-4}$ (at $95%$ c.l.) in units of the number density of CMB photons, perfectly accounts for the difference between predicted and observed $^7$Li primordial abundance. This model also predicts an increase of the effective number of relativistic degrees of freedom at the time of CMB decoupling $Delta N_{eff}^{cmb}equiv N_{eff}^{cmb}-3.046=0.34_{-0.14}^{+0.16}$ at $95%$ c.l.. The required abundance of sterile neutrinos is incompatible with the standard thermal history of the Universe, but could be realized in a low reheating temperature scenario. We provide forecasts for future experiments finding that the combination of measurements from the COrE+ and PIXIE missions will allow to significantly reduce the permitted region for the sterile lifetime and density.
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