Irrevocable Family Trust is a separate legal entity and is required to file a tax return on the form 1041 - http://www.irs.gov/pub/irs-pdf/f1041.pdf if it has any taxable income.
Unless trust documents have any limitations - generally - there is a choice - either having this income taxable to the trust, or
distribute the income to beneficiaries and report each beneficiary's share of taxable income on the schedule K-1 - http://www.irs.gov/pub/irs-pdf/f1041sk1.pdf - in this case it will be excluded from trust taxable income.
You may also distribute to beneficiaries only part of income and have the rest taxable for the trust.
You may choose to have the income being taxable for the trust and distribute to beneficiaries after tax funds or do not distribute at all.
The fact of distribution doesn't eliminate the filing requirements for the trust.