At the risk of turning this into a bikeshedding topic, I’d suggest not showing anything when everything is fine. No reason to bring attention to it if there’s nothing to bring attention to.
For the other states you could use some “in progress” icon if it’s being built and a warning icon if it failed. But more importantly I think is that the tab should be in a disabled state in both cases, and that the icons, or the tab as a whole, should have a tooltip with explanation. Ideally the tooltip would also be shown when the (disabled) tab is clicked, so that it also works on touch interfaces.