Permanent residents pay tax on their worldwide income. It does not matter that the income is from a foreign bank. Just as important is checking the box on Scheduel B of the Form 1040 telling the IRS that the taxpayer has a foreign bank account. This is a hot issue for the IRS. In addition a US Treasurey form 90-22.1 may have to be filed. You can find that form on the IRS website.
You have a few options that a local attorney should be able to help you with. First, since you want to give your brother all rights to the home either (i) you and your brother as the heirs can transfer the home to your brother, or (ii) if it is not advisable for you to make a gift to your brother you should investigate whether you can "disclaim" your share in favor of the house going directly to your brother. Once your brother owns the house he can make his estate plan, usually with either a...