The Moria facility on the Greek island of Lesbos, which previously housed up to 12,000 people, was destroyed overnight on Tuesday after an apparent arson attack by migrants who have long complained about there.
"It's an immense tragedy," UN chief Antonio Guterres told French channel TV5monde in an interview broadcast on Saturday.
"In my opinion the only solution is transferring these refugees to the continent and I hope there will be European solidarity."
The former Portuguese prime minister continued: "You can't expect the country (Greece) on the frontline to resolve everything. There needs to be shared responsibility within the European Union."
Tensions rose on Lesbos on Saturday after hundreds of asylum seekers after a third night of sleeping rough in doorways and by roads.
Police fired teargas when some of them began throwing stones, an AFP reporter at the scene said.
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Efforts in the past to create a quota system for refugees in the EU, which would have seen all members agree to take in migrants from frontline countries such as Greece and Italy, have foundered due to divisions.
Right-wing governments in many member states, particularly in Poland and Hungary, refused to sign up to the scheme.
Ten European Union member states have agreed to take in a total of 400 unaccompanied minors from Lesbos, but rights groups say the response so far has been insufficient.
Guterres welcomed a Franco-German initiative to distribute the minors, but said "we need to go further."
Agencies contributed to this report.
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