How Do I Book a Train Ticket On Italian and French Rail sites Using my Eurail Pass

Yes, hopefully they sent you directions for downloading, booking and retrieving all of your tickets with and without extra seat reservations and/or supplements. Download their app and familiarize yourself with your Eurail booking platform. https://www.eurail.com/en/plan-your-trip/rail-planner-app

Have you already bought the pass? If not, there may be better options.
Have a great trip!

So you don't need a ticket, you need a seat reservation. You might try Rail Europe, if that is where you bought the pass.

If the Berner Oberland Pass will be active on your arrival day, it is good from Domodossola to Spiez or Bern if you take the route via Kandersteg. These trains are direct at 58 past the odd hours (7:58, 9:58, etc) plus one at 6:58 am and one at 6:58 pm. You could take a regional train from Milan to Domodossola that does not need a seat reservation.

You can use all regional trains to get to Beaune by going to Geneva, then Lyon, then Beaune with only 2 connections. About a half hour faster, but requiring 5 connections is a route through Biel, Meroux, Belfort, Besancon, and Dijon. All regional trains with no reservations needed.

If you haven't bought an Eurail Pass yet you have just dodged a bullet.

Last summer my sister was travelling through Europe, and had an Eurail Pass. She tried to get reservations for Milan to Spiez, but was unsuccesful. Even the station in Italy could not help them. Since in Italy long distance trains come with included reservation most ticket clerks don't even know anymore how to make separate reserrvations.
In the end she just bought tickets, as they were in fact cheap.

So just book Milano - Spiez on www.trenitalia.com and Bern - Beaune on www.sbb.ch, and forget about getting a pass.

Well, thank you VERY much everyone for the help. It concerns me that one person said their relative was not able to use the pass between Milan and Spiez, Italian Train conductors did not even know how to ticket them? I was all set to get a 4-day within one month pass for a sale price of $239. It seems a lot cheaper than separate tickets from Milan airport to Cinque Terre (Day 1), Cinque Terre to Domodossolo (Day 2), Basel to Dijon (Day 3), and Dijon to Paris Airport (Day 4). Now not so sure, maybe I will just bok train tickets when I get there...

If your trip is in the fall, you seem well-positioned to purchase tickets shortly after they go on sale (which I imagine will not be for some time yet). That may well cost less than the Eurail pass, and you probably won't have to worry about buying seat reservations, because if a seat reservation is mandatory, it's already included in the price of the point-to-point ticket.

Rail fares can vary a lot from day to day and train (departure time) to train. I randomly chose the date of June 5 for the Italian trips and May 1 for the French trips and looked up the fares for the trips you mentioned. The French rail company doesn't have all its bargain tickets loaded even as far out as June 5.

Milan - Monterosso (CT): €10 - €26
Monterosso - Domodossola: €21 - €54
Basel - Dijon: €44 - €60
Dijon - Paris Gare de Lyon: €13 - €44

Buying fall tickets when they initially go on sale would probably yield better fares than those, and your seat reservations would already be covered where they are needed.

As you can see, even using the highest May 1 / June 5 fare I found for each trip comes out a lot less than €239, which would be a good bit higher by the time you bought the seat reservations.

Thank you again SO MUCH for the additional information. Wow, it is great to have so many experts to help in my first multiple day train excursion across 3 European countries. I will look at those train fares again and I agree now that I will not need a Eurail Pass for what we plan to do.