I'm not aware of any demo instance to play with, but it can be downloaded from: https://www.jenkins.io/download/
LTS version is fine to check rendering issues on Pale Moon (version 2.516.2 at the time of this writing).
It needs either Java 21 or Java 17 to be installed to run, Java 21 (JDK 21) can be downloaded from: https://www.oracle.com/java/technologies/downloads/
You can pick system account during install so you don't need to configure anything additionally. When installation of both is done, it can be accessed on http://localhost:8080/. You have to go through setup wizard, you can let it install suggested plugins and skip creating additional user.
When all is done, first page is login page. I'm not sure how extensive rendering issues are, presumably most are visible on login page.
Anyway, visual comparison of login page between Pale Moon 33.8.1.2 and Firefox 140.2.0esr is below. Firefox 125.0.3 from April 2024 is not enough to render this page correctly, but Chromium 120 from December 2023 is.


And Firefox 125:

Ignore the icon in the username field, it's from KeePassXC extension.


