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The water in the cup appears to be blue under an IR camera because of evaporate cooling–water molecules constantly evaporate and take away thermal energy.
The evaporated water molecules condense onto the underside of the paper and release heat.
The heat conducts through the thin paper and shows up on the upper side.
The heating effect fades because the condensation and evaporation of the thin water lay is reaching equilibrium, resulting in less and less heat release.