How much does the distance between a transcription factor binding site and a promoter influence transcription?
That really depends on your system. At least for yeast the difference influences the strength of the activation ("Analysis of Transcriptional Activation at a Distance in Saccharomyces cerevisiae").

For bacteria such long distance regulations have recently been identified. Before that it was thought that this does happen only in eukaryotes. See the paper: "Long distance relationships: enhancer-promoter communication and dynamic gene transcription."

I eukariotic cells the regulatory sequences (transcription factor binding sites and enhancers as well as silencers) can be really far away. An example for this would be the rs12913832 in intron 86 of HERC2, which influences the expression of the OCA2 gene, which is about 21 kB away. Both genes are not related in their functions. Normally three transcription factors interact here in the regulation of OCA2, HLTF, LEF1 and MITF, which leads to the formation of a DNA loop between the enhancer in HERC2 and the promoter in OCA2. The polymorphism reduces the binding of HLTF to the enhancer and hinders the formation of the loop which results in reduced expression of OCA2 (the whole story can be found here: "HERC2 rs12913832 modulates human pigmentation by attenuating chromatin-loop formation between a long-range enhancer and the OCA2 promoter.")

This paper ("Essential role of microphthalmia transcription factor for DNA replication, mitosis and genomic stability in melanoma.")analyzed (among other things) the binding sites for the microphthalmia-associated transcription factor (MITF) and their distance from the transcription start site. Figure 1 a/b of the paper is shown below. Interestingly a lot of the binding sites where pretty distant. 



To go deeper into the topic, the following reviews are interesting to read:


Long distance relationships: enhancer-promoter communication and
dynamic gene transcription.
Enhancer and promoter interactions-long distance calls.