What can make me ineligible for a work permit as an asylee?

It depends. Generally, the following must be true to qualify for a work permit as an asylee: 

1. You have NOT been convicted of an “aggravated felony.”
2. You have NOT failed to appear for an asylum interview or a hearing before an Immigration Judge (unless the applicant demonstrates exceptional circumstances for having failed to appear). 
3. You have NOT had his asylum application denied by an Asylum Officer or by an Immigration Judge within 150 days after applying for asylum.  [Note: A referral of the asylum applicant to Immigration Court for removal proceedings is not considered a denial of the application and therefore does not prevent issuance of an EAD.]
4. You have NOT asked for a continuance in Immigration Court before 180 days since the filing of the application.