Short answer: No.
The DST changes deal with the Java Virtual Machine [JVM], and your own code that you are running inside
If SE is using an older JVM... one with invalid DST representation, then so is the code that you
are running inside SE (i.e. your webapp code).
You would only care about this if the code that runs inside your webapp does things that rely on the DST
representation being correct.
So it really depends upon the code that you are running inside your Web Application (i.e. inside ServletExec).
Take a look here:
I'd recommend that everyone upgrade their JVM as described at
the link above, and strongly encourage that FAQ #338 be consulted also.
And check the System Requirements for whichever version of SE you
are using to ensure that you upgrade to a JVM that is in fact supported by that SE version.
This issue was also discussed on the SE Interest List,