Several countries have cut ties with Qatar, accusing it of supporting terrorism in the Gulf region.
Qatari nationals in Bahrain, Saudi Arabia and the UAE have been given two weeks to leave.
Qatar denies backing militants and its foreign minister has called for "a dialogue of openness and honesty".
Egypt said it was closing off its airspace to Qatar from 04:00 GMT on Tuesday "until further notice".
Travel disruption is expected as the airport in Doha, Qatar's capital, is a major hub for international flight connections.
Airlines affected will include Qatar Airways, Etihad Airways and Emirates.
Flight tracking technology by the FlightRadar24 website shows the limited route being taken by Qatar Airways flights on Tuesday morning
When avoiding the massive neighbour to the west, Saudi Arabia, Qatari planes will inevitably have to take longer routes leading to longer flight times.
But Qatar's Foreign Minister, Sheikh Mohammed Bin Abdulrahman al-Thani, told broadcaster Al Jazeera the country would "still have access to the world through international sea lanes and international airspace".
An anonymous Somali official told AP news agency at least 15 Qatar Airways flights had used Somalia's airspace on Monday, many more than on a normal day.
Who has done what?
The states who joined the move against Qatar, a tiny but gas-rich peninsula, on Monday include some of the biggest powers in the Arab world.
Saudi Arabia, Bahrain and the UAE closed all transport ties by air, land and sea to Qatar.
They gave all Qatari visitors and residents two weeks to leave their territory, and banned their citizens from travelling to Qatar.
The UAE and Egypt expelled Qatari diplomats, giving them 48 hours to leave.
Saudi Arabia closed down a local office of Al Jazeera but said Qatari citizens would still be allowed to take part in the annual Hajj pilgrimage to Mecca.
Egypt, Yemen, Libya's eastern-based government and the Maldives later followed suit in severing diplomatic ties.
In a country reliant on imported food, residents began to stockpile. AFP news agency said queues in one shop were 25 people deep.
Why has this happened?
While the severing of ties was sudden, it has not come out of the blue, as tensions have been building for years, and particularly in recent weeks.
Broadly, two key factors drove Monday's decision: Qatar's ties to Islamist groups, and to Iran, Saudi Arabia's regional rival.
Wealthy individuals in Qatar are believed to have made donations and the government has given money and weapons to hardline Islamist groups in Syria.
The Financial Times also reports that Gulf allies were angry that Qatar paid a $1bn (£773m) ransom to jihadists and Iranian security officials after Qatari nationals were kidnapped in Iraq and Syria.
Analysts also say the timing of the diplomatic withdrawal, two weeks after a visit to Riyadh by US President Donald Trump, is crucial.
Mr Trump's speech in Saudi Arabia, in which he blamed Iran for instability in the Middle East and urged Muslim countries to take the lead in combating radicalisation, is likely to have emboldened Gulf allies to act against Qatar.
In the same week as Mr Trump's speech, Egypt, Saudi Arabia, Bahrain and the UAE blocked Qatari news sites, including Al Jazeera. Comments purportedly by Qatari Emir, Sheikh Tamim bin Hamad al-Thani, criticising Saudi Arabia had appeared on Qatari state media.
The government in Doha dismissed the comments as fake, attributing the report to a "shameful cybercrime".
What has been the reaction?
Qatar, which is due to host the football World Cup in 2022, said the decision would "not affect the normal lives of citizens and residents".
Iran, Turkey and the US have called on all sides to resolve their differences. Kuwait has offered to mediate talks.
Sudan's foreign ministry also called for dialogue and said the country was "fully ready to undertake all efforts in order to achieve calm and reconciliation that would help serve the interests of the people of the region".
Qatar's stock market closed down 7.27% on Monday.