There were apparently no laws for asylum seekers in any Gulf countries until autumn 2018.
Qatar: Gulf’s First Refugee Asylum Law [...]
Qatar passed Law No. 11/2018 on Organizing Political Asylum on September 4, 2018, alongside two other laws regulating residency in the country. One abolished exit permits for most migrant workers, and the other allows people for the first time to apply for permanent residency. Both point to Qatar’s increased respect for international standards, but, as with the new asylum law, neither fully aligns with international human rights law.
And as the HRW article goes on to detail, there are quite a few caveats in the Qatar asylum law. And by Feb 2019, they had not accepted anyone under that law:
The government has so far failed to implement a law passed in September 2018 that sets out the standards for granting asylum and the rights and benefits for people granted asylum in the country. The government says that the infrastructure necessary to enforce the law has not yet been set up.
Outside of Gulf countries per se, Turkey (as you probably know) has taken in quite a few Syrian refugees, but their legal status is rather recently decreed and as with many other things in Turkey recently, the legal facade is sometimes not that relevant in practice.
In 2013, Turkey adopted its own legal framework on the protection of asylum seekers and refugees. In October 2014, Turkey also adopted a regulation under which it grants Syrians temporary protection. As of June 28, 2018, Turkey said it had registered 3,562,523 people under the regulation. Registered Syrians are entitled to assistance. Even though the regulation says Syrians who fail to register will not be deported to Syria and will only face an “administrative fine,” Human Rights Watch found that unregistered Syrians have been deported for not having temporary protection permits. [...]
Agencies said their extremely limited contact with unregistered Syrians means they can neither estimate how many unregistered Syrians now live in Hatay and other [Turkish] provinces, nor the extent to which the registration suspension has led to deportation and denial of service access.
And regarding Lebanon,
There are approximately 1.5 million Syrian refugees in Lebanon; however, 74 percent lack legal status. Authorities heightened calls for the return of refugees in 2018 and municipalities have forcibly evicted thousands of refugees.
And even those that do have legal status, what they seem to have is a "circulation permit" that entitles them for a 12-month stay in Lebanon while UNHCR is trying to resettle them in other countries.