Learn Hardcore Japanese: huta wo aketa mama de ha, misosiru ga sugu sameru kara, saki ni tyawan wo youisite. - If you leave the lid open, the miso soup gets cold quickly, so get the bowls ready first.

Breakdown of futa o aketa mama de wa, misoshiru ga sugu sameru kara, saki ni chawan o youishite.

What does 開けたまま mean here, and why is 開けた in the past form?

開けたまま means left open or in the state of having opened it.

The た-form here does not mean past time in the usual sense. In the pattern V-たまま, it shows a resulting state:

So フタを開けたまま means with the lid left open, not necessarily after you opened it in the past.

What is では doing in 開けたままでは?

Here, では adds the sense of if it stays in that state or in that case.

So:

You can think of it as setting up a condition based on a state.

The whole chunk フタを開けたままでは means something like:

Why is 冷める used instead of 冷える?