Measuring spectral distortions of the cosmic microwave background (CMB) is attracting considerable attention as a probe of high energy particle physics in the cosmological context, since primordial inflation explorer (PIXIE) and polarized radiation imaging and spectroscopy mission (PRISM) have recently been proposed. In this paper, CMB distortions due to resonant conversions between CMB photons and light axionlike particles (ALPs) are investigated, motivated by the string axiverse scenario which suggests the presence of a plenitude of light axion particles. The distortions due to resonant conversions have a frequency-independent shape, in contrast to μ and y distortions. Therefore, one can distinguish the distortions due to resonant conversions from μ and y distortions. Since these resonant conversions depend on the strength of primordial magnetic fields, constraints on CMB distortions can provide an upper limit on the product of the photon-ALP coupling constant g and the comoving strength of primordial magnetic fields B. Potentially feasible constraints from PIXIE/PRISM can set a limit gB ≲^10-16 GeV-¹ nG for ALP mass m_ϕ≲^10-14 eV. Although this result is not a direct constraint on g and B, it is significantly tighter than the product of the current upper limits on g and B.