What are the effects of removing CD4 receptors?
The CD4 receptor is vital for the proper functioning of the immune system. It is found not only on T-Lymphocytes, but also on macrophages and dendritic cells. Its function on T-cells is to stabilize the interaction between the T-Cell receptor and the MHC Class 2 (often known as HLA II in humans) antigen complex on antigen presenting cells and improves the affinity dramatically. 

CD4 is also vital for the activation of Helper T-Cells (2). It is needed in the signaling which occurs in the thymus that selects for CD4+ T-Cells. If you were to delete CD4, you are, in effect destroying the adaptive immune system in humans, as there would be no CD4+ produced. In this situation, you would not get activation of B-Cells or Cytotoxic T-Cells, and the immune system would only be left with its innate defenses to protect it. This is found in diseases such as Severe Combined Immunodeficiency and can only be cured with early bone-marrow transplantation.

A better target would be the CCR5 receptor. This receptor acts as a coreceptor for the interaction between GP120, the receptor on the surface of the HIV capsid and the CD4 receptor. People who have a mutation in the CCR5 receptor are resistant to the most prevalent isoform of HIV, as the interaction is necessary for the capsid to fuse with the plasma membrane of the CD4+ cell.

See this article about the CCR5-∆32 mutation. HIV Resistant Mutation