The teenager, who was arrested on Wednesday night, is denying any guilt.
"He is currently indicted for aggravated criminal damage, but I can not rule out that the indictment will be expanded," Ole Bjørn Sakrisvold, of the Vestfold police told VG.
Fourteen people were hospitalised when an apartment block was set alight in the city on Tuesday night.
Sakrisvold said police could link the man arrested to the apartment fire but had yet to link him to a further two fires which took place at the same time, one in a boat and one at a mobile home.
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Tønsberg, around 60 miles outside Oslo, is regarded as Norway's oldest town.