At this point in time, there is no general ability to disable banner ads (such as "Instance Health - Now Available in Portfolio Insights ...")that pop up in Jira Datacenter, and this request is to have the ability to do so in a single, unified place.

Current Workarounds (that incompletely address the issue):

A number of banners come from the Atlassian Troubleshooting and Support Tools(ATST) plugin, some are product ads and some are healthcheck alert banners. You can disable these popups by:
1) Going to Admin (⚙) > Manage Apps > Manage Apps
2) Searching for Atlassian Troubleshooting and Support Tools
3) Expand the section that says X of X modules enabled
4) Searching the module list for Troubleshooting Resources Healthchecks Notifications (atst-healthchecks-notification)
5) Disable the module

The healthcheck capabilities will still be enabled and visible via Admin (⚙) > System > Troubleshooting and Support Tools > Instance Health tab, but you'll no longer receive popups about them

Additionally, Jira also has a "newsletter tip" that pops up with various suggestions, some of which can include ads. To disable this pop-up:
1) Go to Admin (⚙) > System > General Configuration
2) In the top right corner, click on Advanced Settings
3) Find the key labeled jira.newsletter.tip.delay.days
4) Click the pencil to the right of the key value pair and change the value to -1 to disable it