Is oxygen necessary for burning?
Yes, absolutely. Burning refers to oxidation accompanied by a flame and there are numerous reactions in which oxygen takes no part. Most of these are either with elementary halogens, or compounds thereof. 

Here are some examples.

\begin{align}
\ce{2K + F2 &amp;-&gt; 2KF}\\ 
\ce{2Fe_\text{wool} + 3Cl2 &amp;-&gt; 2FeCl3}\\ 
\ce{H2O + ClF3 &amp;-&gt; HF + HCl + OF2}\\ 
\ce{2H2O + ClF3 &amp;-&gt; 3HF + HCl + O2}\\ 
\ce{2Cl2 + 2OF2 &amp;-&gt; 4ClF + O2}
\end{align}

So in fluorine chemistry, you don't oxygenise it; it flourises you.