* Qatar Insurance Co says it can't renew Abu Dhabi licence
* Initially, short-term funds pulled out of Qatar
* Now long-term business links in region being affected
* Old links may not revive even if dispute resolved
* Qatar can continue operating but with higher costs (Writes through with context)
DUBAI, Sept 18 (Reuters) - A Qatari insurance company is closing its branch in Abu Dhabi because authorities there have not renewed its licence, a sign of how the region's diplomatic crisis is starting to have long-term effects on its economy.
Qatar's stock index also hit a five-year low on Monday because of concerns over the impact of the diplomatic dispute on some of its companies.
Qatar Insurance Co said on Monday its Abu Dhabi branch, open since 2002, would shut because it could not renew its licence. In response, the firm's share price dropped 2.3 percent; the branch earned gross premiums of about $30 million annually.
In the initial weeks after the United Arab Emirates, Saudi Arabia, Bahrain and Egypt cut diplomatic and transport ties with Doha on June 5, accusing it of backing terrorism, many long-term investors sat tight, hoping the dispute would be resolved soon and allow normal business to resume.
Gulf investors withdrew deposits from Qatari banks and pulled funds out of Doha's stock market, but with the exception of Qatar Airways, Qatari companies continued to operate in the four Arab states.
Now, however, there are indications that the damage to business ties may last for many years, as Qatari firms begin to close operations and sell assets - steps that will be hard to reverse even if the diplomatic crisis ends.
"The pull-back by Qatar Insurance is another sign that the fraying of political ties between Qatar and its Gulf neighbours is also damaging economic ties," said Jason Tuvey, Middle East economist at Capital Economics in London.
"The move suggests there is little confidence that relations will improve any time soon, and we could see more Qatari corporates downsize or withdraw their operations in the rest of the region."
Because of Qatar's vast wealth - it is the world's largest natural gas exporter - it is able to cope with the isolation. But it will have to pay higher costs, many of them borne by the government, to do business with the rest of the world, said another analyst monitoring the region.
Doha Bank, Qatar's fifth-biggest lender, has cut around 10 jobs in the UAE and plans to put some staff in the region on unpaid leave, sources told Reuters last week. Its UAE operation has been in talks to try to sell some of its assets.
Last month, major shipping and logistics group Qatar Navigation said it was shifting its regional hub from Dubai to the Omani port of Sohar because direct shipments between Jebel Ali and Doha were no longer possible.
Companies in the UAE, Saudi Arabia and Bahrain have not so far announced formal exits from Qatar, but Qatari trade statistics suggest their exports to Doha have plunged since the crisis began.
They may also be starting to sell assets in Qatar. A company owned by a member of the Abu Dhabi royal family is seeking to sell Dolphin Tower in Doha, the Qatari headquarters of natural gas supplier Dolphin Energy, sources told Reuters.
Previously, many multinational companies in the region used Dubai as a base to do business with Qatar. But because of disruption to shipping and banking links, some are now handling Qatari business out of other centres such as London - a more expensive and time-consuming alternative.
An international banker in the Gulf, who declined to be named because of political sensitivities, said the rift between Qatar and its neighbours appeared to be widening as the dispute dragged on.
If the crisis worsens, other countries could formally sever financial ties with Qatar, which would deepen Doha's isolation, he said.
At the moment there are no credit or capital controls, but that is the next one to watch out for." (Writing by Andrew Torchia; Editing by Keith Weir)