It's a good idea to apply for your green card as soon as you can, for two reasons:

• Your continuing status as an asylee depends on the conditions in your home country. If the U.S. government decides that Burma's political situation has improved, it could conclude that you and your family are no longer in danger and take away your asylum status. This doesn't happen to asylees often, but it's a risk worth avoiding.
• Getting a green card is a necessary step toward U.S. citizenship. The most secure status you can have in the United States is citizenship. It will allow you to vote, sponsor other family members for immigration (such as your parents and adult or married children and siblings), and qualify you for certain federal jobs. Being a citizen also protects you from the possibility (faced by green card holders) that you will be removed (deported) for having spent too long outside the United States, been convicted of certain crimes, or violated the immigration laws. But you need to spend five years as a permanent resident (with a green card) before you can apply for U.S. citizenship. Luckily, the year you spend as an asylee counts as one of the five years.

If you truly can't afford the application fee for a green card, you can ask for a fee waiver when you send in your application. (As an asylee, you're lucky that the U.S. government doesn't require you to show that you are financially self-supporting before approving your green card -- a requirement for most other green card applicants.) USCIS provides instructions for requesting a fee waiver on its website, here.