No, receiving the 2021 Child Tax Credit or advance Child Tax Credit payments will not affect your immigration status or ability to get a green card. Under current law, receiving the Child Tax Credit or other Federal tax credits that you are eligible for will not impact your immigration status, your ability to get a green card, or your future eligibility for immigration benefits. The use of Federal tax credits is not considered for purposes of a "public charge" determination by U.S. Citizenship and Immigration Services.
Sources:
Publication 5534-B (06/2021)
Publication 971 (12/2021)
Publication 5534 (06/2021)