Non-Schengen country - Will this extend 90 day visa?

Will he have a student visa for Italy?

You can try using the Schengen visa calculator to see

https://www.visa-calculator.com/

This program did not provide student visas. It was designed to be 90 days only.

The answer is yes, any days he spends outside of Italy in Ireland will add to his ability to stay after his program ends. I suggest he spend as much time as possible in the UK on spring break to save days. The rule is 90 total days in a 180 day period which starts the day he arrives in Italy.

I suggest he spend as much time as possible in the UK

Is he going to the UK or the Republic of Ireland?

either would work, as would any other non-Schengen country

but if he goes to the UK he will need an ETA

Any part of a day in the Schengen, means a day used, even if an hour or less. So in your original question, he is spending 5 days in Ireland, but since his arrival day, and his departure day will be partly in the Schengen zone, Yes, he only gains three days in the Schengen area.

Ireland and the UK are easy options if he has other time he can spend out, otherwise he might look to the Balkans, Montenegro, Albania, both close, even Bosnia and Serbia would be options.

You are correct. It is always a look back every day. Where has he been been for the past 180 days. You have only 90 days in the prior 180 so counting carefully is important and you appear to be very aware of the situation. Now if he does do a few days out of the schengen zone, he needs to be sure of passport stamps or other documentation since it is so tight. Good luck