Both of the previous answers didn't help me as one didn't work on non-delegated tooltips, whereas with the other method I could not play around with CSS in the dev panel. The tooltip did stay open, but I could not uncheck any checkboxes against any CSS like I would normally or type in more CSS - to get to a solution before implementing it.
Ultimately I used the debugger statement to break in my code after tooltip was opened.
Example:
jQuery(".myElement").tooltip({
open: function (event, ui) {
debugger;
}
});
When that breakpoint is hit, evaluate this in the console:
$(".ui-tooltip-content")
And then right click on results of the evaluation in console and choose Reveal in Elements panel to work with it just like any other element on the page.
This is especially helpful when tooltip content is constructed dynamically (like in my case).