As part of our tour of Italy we will visit Puglia, headquartering ourselves for five days in Lecce. From Lecce, we intend to travel north, probably to Umbria, preferably on Friday September 20. The Italian train website shows night trains from Lecce north to either Bologna or Rome Termini to be sold out. Is this likely correct, or is it possible that they just haven't opened up reservations yet? I see only one date on which I could purchase tickets. Do these seats really sell out 4 months in advance?

given that there are still a few spaces left in the coming week I suspect you are just too early

There are 2 ICN Lecce Milano and 1 ICN Lecce - Torino every night.
I seems in fact that the Friday night trains are sold out.
On all other days of the week, 2 bed cabins or couchettes are available.

But I don't understand why you want to reach Ancona at 3am in order to change there to the 3.33 am regional train to Foligno and probably wait there until the 5.57 IC brings you into the still sleeping cities of Assisi or Perugia.
IMO it would be way more relaxing to leave Lecce at 12.57 and to arrive at Foligno/Assisi/Perugia at time for dinner.
Alternative: Lecce dp 17.17 - sleep at Rome (close to Terrmini station) - Perugia ar 10.17

Does it have to be a night train?

I searched Lecce to Perugia which, as you found, is supposedly "sold out" for the first leg (Lecce to Rome). However, this route involves two connections. The first is in Ancona at 3:09 a.m. (24 minutes) and the second in Foligno at 5:20 a.m. (10 minutes). The overall journey takes 9:48 hours, arriving Perugia at 6:13 a.m. Only the first train (Lecce to Rome) offers sleeping cars, but I don't think there would be much sleeping that night.

I agree with neckervd. Take a day train.

We decided to take a day train to Napoli. Problem solved.

So you don't want to go to Umbria any longer and chose a completely other destination: Campania.
In order to go there, the direct Bari - Naples buses are way faster (and cheaper) than the train.

We have a Eurail pass and so will take the train. Umbria after Naples.

You are right. The reservation fee for the Frecciarcento for Eurailpassholders is not higher than the full bus fare Bari - Naples (it's about the same amount).