I would split this question into three parts:
Q1. Provided I use the open source license (GPLv3) version of ExtJS in a public facing web application, does this imply any consequenses on my application?
Q2. Which parts of my web application are affected by the GPLv3?
Q3. For the parts of my web application that are affected by the GPLv3, what does that mean?
A1. Depends on your definition of "use". You can use and modify GPL software without any restrictions, as long as you don't distribute it. In the case of a Javascript library like ExtJS, your kind of usage certainly implies distribution in the terms of the GPL. The borderline is whether the code of the GPL software is run internally on the web server, or whether it's distributed to web clients.
A3. You have to release the affected parts under a license compatible with the GPLv3. Besides other requirements, you have to make clear that the code is licensed under the terms of the GPLv3 (or compatible license), and you have to make the source available to anyone who asks. The exact requirements are written down in the text of the GPL.
A2. This is the messy part of the question, since appearently different people have very different opinions about this. The ExtJS people appearently believe that every part of the web application has to be put under a license compatible with the GPLv3. Many other people have quite contrary opinions where to draw the borderline.
For C code, the classical example was linking against a GPL library (GPL is invoked) against calling a GPL program (GPL is not invoked). Make up your mind how this applies in the case of ExtJS.
The official FAQ about the GPL has an example about client-side JavaScript.
My personal conlusion: Respect the intent of the authors of ExtJS. If you're building on their work, then play with their rules and don't try to play tricks. If their ideas about licensing are not acceptable for you, then look for another library with a more acceptable license.