The FU Ori-type young stellar objects are characterized by a sudden increase in luminosity by 1-2 orders of magnitude followed by a slow return to the pre-outburst state on timescales of ̃10-100 yr. The outburst strongly affects the entire disk, changing its thermal structure and radiation field. In this paper, using a detailed physical-chemical model, we study the impact of the FU Ori outburst on the disk chemical inventory. Our main goal is to identify gas-phase molecular tracers of the outburst activity that could be observed after the outburst with modern telescopes such as ALMA and NOEMA. We find that the majority of molecules experience a considerable increase in total disk gas-phase abundances due to the outburst, mainly due to the sublimation of their ices. Their return to the pre-outburst chemical state takes different amounts of time, from nearly instantaneous to very long. Among the former, we identify CO, NH3, C2H6, C3H4, etc. Their abundance evolution tightly follows the luminosity curve. For CO, the abundance increase does not exceed an order of magnitude, while for other tracers, the abundances increase by 2-5 orders of magnitude. Other molecules, like H2CO and NH2OH, have longer retention timescales, remaining in the gas phase for ̃10-103 yr after the end of the outburst. Thus, H2CO could be used as an indicator of the previous outbursts in the post-outburst FU Ori systems. We investigate the corresponding time-dependent chemistry in detail and present the most favorable transitions and ALMA configurations for future observations.