I can’t nail down a set of repro steps to get into this state, but I’ve seen this happen multiple times over the last 6 months, in both Chrome and Firefox.
If the browser-based tooltip appears for an object that can be dragged, it blocks the dragging interaction. I’ve seen it affect the projects list under a Team in the sidebar, as well as a task inside a project. The latter can be seen below in the embedded gif.
Notice that on click interaction and hold, the UI doesn’t reflect a drag is happening. Unfortunately, all dragging interactions are blocked (not just dragging the object with the active tooltip), and this never seems to fix itself, including after refreshing the page. The only “workaround” is to open a new tab.