Yes, you do need to book a ticket for a specific tour date and time. You don't necessarily need to book in advance, as they keep the ticket sales open right up to, and including, the day of the tour, but they can sell out for a given time or date. You can get tickets here:

https://www.mlb.com/redsox/ballpark/tours

Normally, I'd tell you that for a random Monday you'd be fine waiting until the day-of to book whatever time suits you, but given that the Monday in question here is Memorial Day, that won't be the case. I think it's very likely that most or all of the tours that day will sell out by the time that day actually rolls around. If you don't want to commit to a timeslot yet then I'd say just keep an eye on the ticket sales website and see when some of the timeslots start disappearing from the list and determine then if you're able to commit to one at that point.