How can I make kittens-game run while not using that tab?

Many browser games do not run while you're in another tab. While Kittens Game has a "web worker" option to allow it work in inactive tabs, I've had poor results with it. Instead, a work-around I've often used to great effect is to run the game in a separate window. In Firefox, this can be easily done by right clicking the tab and choosing Move to New Window. Other browsers should work very similarly.

Click on Options and select Use web worker (game works correctly in background tab, may cause performance issues)

Depending on what browsers you have available another option would be to run the game in a separate window, or failing that a browser other than the one you normally use. So if you usually use Firefox, having it running in Chrome it should still progress even if Chrome is minimized.

If you do decided to run the game in another browser, you should be able to export your game and load it using the Export and Import functions from the Options menu.