While excellent thermal and mechanical stabilities of all-polymer solar cells (all-PSC) have been demonstrated, the storage stability of all-PSCs has rarely been studied. In this paper, the storage stability of all-PSCs is systematically investigated and compared to fullerene-based polymer solar cells (PCBM-PSCs). We identify that the efficient inverted type all-PSCs made with a molybdenum oxide (MoO3) anode interfacial layer exhibit degradation over short periods of storage even under inert nitrogen-filled and dark conditions, while the control inverted PCBM-PSCs containing the same polymer donor are relatively more stable. To elucidate the origin of the poor storage stability, morphological and electrical properties of all-PSCs are investigated. We reveal that the work function of MoO3 is largely changed during the storage because of the interaction between MoO3 and the underneath naphthalene dimide (NDI)-based polymer acceptors (P(A)s). This causes unfavorable energy-level alignment in devices, resulting in increased charge recombination and deteriorated charge collecting efficiency. To resolve this issue, we propose two effective strategies: (i) introducing a passivation layer to physically separate the NDI-based P(A)s and MoO3, and (ii) replacing MoO3 with an efficient polymer interlayer. We prove that the modified all-PSCs not only exhibit excellent storage stability with high power conversion efficiency for more than 45 days, but also show high air-stability even without encapsulation. Our findings provide deeper understanding of the storage stability of all-PSCs and suggest future guidelines for efficient and burn-in free all-PSCs.