You need an embassy visa for Tonga if you have an Icelandic passport
Visa options for Tonga
Visitor visa
Travellers need a visa to visit Tonga for Business or Tourism. A paper or embassy visa is the only visa available for travellers visiting Tonga. The visa is typically a stamp or sticker added to the passport. Travellers must obtain this visa from a local embassy, consulate or visa center before travelling.