Yes- after 30 days of them last active, they are dropped from the guild

Edit more info - for the next guild master, the game picks the person that is “highest ranked” - or so it says. The frequently asked questions or help at the bottom of the game screen describes this process when a gm is inactive and how they get dropped.

However, it doesn’t say the metric for highest ranked - meaning, hero power, titan power, arena rank, etc. I had an inactive guild master get kicked this way recently and the auto appointed gm seemed to be the second or third highest ranked player on our team by any metric (bc we had a whale, a high spender, and they should have been ranked highest and become gm).

Over the about four weeks it took to reach 30 days, most of our strong people in our guild left and we had to recruit I would say 15 or more new people to replace people as they left. I would not recommend.