Spending the nights of the 11th, 12th (and optionally 13th) in منى is واجب; being present for the majority of each night fulfills the requirement, and missing it without excuse requires فدية.
Yes, staying overnight in منى during the Days of التشريق is considered واجب (واجب) by the majority of scholars. Specifically, الحجاج should spend the nights of the 11th and 12th of ذو الحجة in منى, with the 13th night being optional (as early departure on the 12th is permitted). The requirement is to be present in منى for the majority of the night — roughly from Maghrib (sunset) to Fajr (dawn).
The tent cities of منى are organized by country and الحج group, with assigned camps. Life in the منى tents is communal and simple — الحجاج sleep in air-conditioned tents, share meals, and use their time between stoning rituals for worship, القرآن recitation, and rest. The days in منى are spiritually significant as they are the Days of التشريق, during which النبي (صلى الله عليه وسلم) said: 'The Days of التشريق are days of eating, drinking, and remembrance of Allah.'
If a الحاج does not stay overnight in منى without a valid excuse, a فدية (sacrifice of a sheep) is required عند جمهور العلماء. Valid excuses include: genuine inability to find space in منى (which does occur due to overcrowding), medical emergencies, or being assigned accommodation outside منى boundaries by the authorities. Those whose camps are technically outside منى's boundaries should still spend as much time as possible within the boundaries.
النبي (صلى الله عليه وسلم) said: 'The Days of التشريق are days of eating, drinking, and remembrance of Allah Most High.'
النبي (صلى الله عليه وسلم) said: 'The Days of التشريق are days of eating, drinking, and remembrance of Allah Most High.'
Source: ابن قدامة, Al-Mughni; al-Nawawi, Al-Majmu
النبي (صلى الله عليه وسلم) said: 'The Days of التشريق are days of eating, drinking, and remembrance of Allah Most High.'
Sahih Muslim #1141 (sahih)