No, Yes, No, Absolutely NOT. The burn marks are forever, they can't be removed so you should change it if you're going to wear it. As Rob stated there really is no danger in wearing it. If you are scared and consider burying it, please consider burying it in my safe, not your back yard. Have it serviced, have the dial repainted and relumed - don't forget to have the hands relumed as well (don't worry they won't use radium paint this time) then WEAR IT, proudly, it's a great watch.
I've been involved with a series of online discussions on Radium since my days at watchmaking school.
We had members with Geiger counters test the actual emissions of the dials, and did sweeps of areas used to service or re-paint the dials after removing the old Radium paint. It has been an exhaustive couple of years, and this is what we discovered:
There is little risk when wearing the watch, the main danger is from ingestion. We do not recommend anyone attempt to re-finish the dials without proper prep or training. Any tools used will be contaminated and showed small amounts of radioactivity after being used. The bench, work pad, and other areas showed signs of contamination, even with precautions taken.
A lot has been published and even movies made regarding the Radium industry, so I will spare you any long winded history. The main problem is with accidental ingestion, which includes inhaling Radium paint dust. Simply wearing the watch poses a very low risk. In comparison, some areas we live in have higher background radiation that what you would be exposed to wearing this watch.
The crystal will have to be replaced, those burns don't buff out. This happens with the old type of celluloid crystal, and as far as I'm aware, doesn't happen with acrylic crystals.
The new lume, as Geoff points out, is non-Radium and not dangerous. The original dial, will remain radioactive even if cleaned. We came to the conclusion that it wasn't worth the hassle of re-luming the old dials, but that is because we come across this situation a lot in our daily work.
More than you ever wanted to know:
http://mb.nawcc.org/showthread.php?70060-a-little-more-on-Radium&highli…
And here is a thread dealing with the whole subject of Radium watch dials, including actual Geiger counter results:
http://forums.watchuseek.com/f11/how-dangerous-501090-post3672724.html
This is a personal decision, and it should be an informed one. My personal opinion is that the risk is low, but I personally have had many radioactive exposures in the form of CT scans, so any additional radiation I am exposed to I take very seriously.
Thank you for your comments and the interesting information.
I do not think I'll be wearing the watch, but I'll probably change the crystal, if I find one that fits.
I'll probably leave the dial and hands in their original state, hoping that will not look too bad, I'm not a big fan of re-finished dials.