Are there any command for producing the BibTeX logo?

It used to be the case that the dtk-logos package provided commands such as `\BibTeX'. It is part of the dtk package.

\documentclass{article}
\usepackage{dtk-logos}
\begin{document}
\BibTeX  % Doesn't work in TeXLive 2015
\end{document}

However, dtk-logos is still under development, and subject to change. A safer option is to use the hologo package.

\documentclass{article}
\usepackage{hologo}
\begin{document}
\hologo{BibTeX}
\end{document}

In fact, the current version of dtk-logos loads hologo, but I think it's probably best to load hologo directly, until development of dtk-logos is complete.

I don't think there's an official BibTeX logo. Usually I adopt \textsc{Bib}\TeX, but also \textsc{Bib}\negthinspace\TeX might be used. The $\mathrm{B\scriptstyle IB}\!$\TeX trick might be used if the font doesn't have small caps.

FWIW, Oren Patashnik's original BibTeX documentation defines a \BibTeX command in the preamble of the corresponding .tex file:

\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

The larger title version is hand-crafted like this:

\title{B\kern-.05em{\large I}\kern-.025em{\large B}\kern-.08em\TeX ing}