Nothing closes on its own when the offer deadline passes: you can carry on comparing offers, accepting one, or using Request revision to have a supplier update theirs.
What changes is the signal on screen. The countdown on the demand page (Closes …) becomes a red Expired tag, and in My Demands the date is shown as past. The status stays Published — MarbleMap has no separate "expired" status for demands.
Ending the demand is your call: Close demand finishes it without awarding, accepting an offer completes it, and Remove from marketplace pulls it off the public list. If you don't want any new offers arriving after the deadline, take that closing step.
Leave it alone and the demand drops off the Marketplace list by itself roughly a month after the deadline. The offers and messages stay under My Demands.
Also asked as: My demand's deadline has passed, what now? · Does a demand close automatically once the offer deadline is over? · Can I still accept offers after the closing date? · The countdown on my demand ran out — what happens to the offers?