Interacting selectively and non-covalently with a Toll-Interleukin receptor (TIR) domain of a protein. The TIR domain is an intracellular 200 residue domain that is found in the Toll protein, the interleukin-1 receptor (IL-1R), and MyD88; it contains three highly-conserved regions, and mediates protein-protein interactions between the Toll-like receptors (TLRs) and signal-transduction components.