I am a US asylee, and I have green card and USA travel document. I want to travel to UAE , and since UAE doesn't accept travel document can I use my national passport?
If you are still a citizen of your country or origjn you can still technically use its passport.
However you should be aware that if you have asylum in the United States that is because you have convinced the US that your country of origin is persecuting you and that you need to be protected from them. Using a passport supplied by them could be seen as putting yourself back under the protection of the people persecuting you. This could result in loss of your asylum status, including loss of your green card.
You should not use your original passport unless you are willing to lose your asylum status. If you absolutely have to go somewhere that needs it, consult an immigration lawyer before you do so.