To be precise, oxygen itself does not burn. It is however needed for combustion. (no oxygen = no flame)
In a monoplace HBO chamber you are under pressure in a 100% oxygen atmosphere; anything that could readily burn is a no no. Anything that could create a spark would have a catastrophic effect. This is why there are the antistatic procedures (wrist strap; and cotton as opposed to synthetics ). Paper is readily oxidized (see the yellowing books in a library), so it would burn easily and "could" also generate static electricity (sparks). They also do not want you to use perfume nail polish and oily substances.... If you have ever seen what happens when you use a greased valve on an oxygen tank you understand why. (It burns like ...hell). This is why O2 tanks have special valves. I am not entirely sure why you need to remove jewelry, gold is very inert. On the other hand jewelry may be composed of many metals and some may act as a catalyst plus there are also watches etc to consider that may be electronic and create sparks.

The above applies for a monoplace chamber where you are surrounded by pressurized 100% oxygen. The situation is different when you are in a multiplace chamber. These are not 100% oxygen and you wear a mask/hood to get 100% oxygen. In such an chamber the requirements should be are lot less stringent.

Hope that makes things a tad clearer.

M