Behavior
If you have deactivated third party cookie support in your browser, the normal function of the app should not be affected. But initially, when you log in to SharePoint, you will see the web part like in the picture below for a short moment. After a while an automatic refresh occurs. Depending on the browser, this effect can also create multiple refreshes until the contents of the web part become visible. The error message thrown has following text: It seems 3rd party cookies are disabled in your browser. Please enable 3rd party cookies for the url {sharepointSite} to use our app.
Solution
The simplest solution is to activate third party cookie support in your browser. You can find out how you can do this here: How to enable 3rd Party Cookies
In case you don't want to allow third party cookies in general it would also be an option to just whitelist the following domains:
[*.]atlassian.net (it would also be possible to just whitelist your Jira Cloud domain directly)