Why do dendritic cells have CD4 or CD8 antigens on their surface? What is their function without the presence of a T-cell receptor?
There is no direct connection between CD (cluster of differentiation) receptors and T-cell receptor (TCR).
CD-receptors are used to label and distinguish different cells belonging to immune system: macrophages, T- and B-cells etc. Dendtritic cells play a significant role as antigen presenting cells and belong to vertebrate (and human) immune system and thus bear certain CD receptor.
These cells participate in the process of learning a new antigen by the immune system. During the very first phase the antigen is taken up by these cells, cleaved and processed. Depending upon the antigenic properties of the molecule it can be recognized as a valid antigen. In this case dentritic cells participate in the process of antigen presentation, where this new antigen bound to the MHC receptor is expressed on the cell membrane.
TCR is the protein that binds to this complex (MHC + antigen) and as long as this binding takes place the signal about antigen is propagated further to the immune system (leading to formation or augmentation of immune response depending upon the cell type which connects to this complex).
I believe rather than canonical involvement in T cell activation the question was what role does CD4 and CD8 have in a non-T cell population. Here is an article that provides some interesting insight: CD4 and CD8: an inside-out coreceptor model for innate immune cells http://www.jleukbio.org/content/86/2/251.full