Yes, our code does make use of Double.parseDouble and the similar Double.valueOf. Unfortunately, the fix is not so simple though, as (according to the article you supplied):
1) a range of numbers trigger the bug
It is not only hanging for 2.2250738585072012e-308.
It is hanging for any number between
2^(-1022) – 2^(-1075) and
2^(-1022) – 2^(-1076).
For example, 2.22507385850720117e-308
2) modified forms of the same number triggers the bug
0.00022250738585072012e-304 (decimal point placement)
00000000002.2250738585072012e-308 (leading zeros)
2.225073858507201200000e-308 (trailing zeros)
2.2250738585072012e-00308 (leading zeros in the exponent)
2.2250738585072012997800001e-308 (superfluous digits beyond digit 17)
So to catch the offending numbers it seems we would almost have to write our own floating point parser 
Still, if there is a straight-forward and performant way to catch these we could consider adding it to LocalUtil.simpleConvert() where our double conversion takes place, with a note to remove it once we expect the world to have upgraded to newer JREs.
Thanks for notifying us of this but I am not seeing how this is a DWR bug. I see how it can affect web applications using DWR and I see that by wrapping Double we could get around it but ....
What are your suggestions?