do you need a reservation for swiss trains - and do the prices change as the travel date approaches?

For Chamonix to Munich there are several routes. But they will all involve the Zurich to Munich train, and for that train I would suggest you get a reservation. In fact, in summer reservations are compulsory on that train.
For the part till Zurich: Most of that will be on trains where you cannot even reserve seats. And yes, Europeans do not reserve seats on local trains.
Switzerland is a small country and the Swiss Railway network is basically a nationwide mass transit system. Think about commuter railways, not long distance.

You can book Chamonix to Munich on www.sbb.ch, and yes, if your date is set do book in advance, as this will be cheaper than buying your ticket on the day itself. When is your trip? If you are trying to book this several months in advance and get funny results or errors you are just to early.

The direct train from Munich to Zurich will have discounted prices on offer if you buy in advance. I checked the price for the full journey on Wednesday March 20, versus travel dates of May 5 and July 5. The entire journey from Munich to Chamonix has a regular full fare of 202-207 CHF, but has a Saver fare of 172 CHF for Wednesday if bought online today. The price is 152 CHF if bought online today for May 5, and 98 CHF for July 5.

In other words, the ticket bought 3.5 months ahead saves you more than 100 CHF over the regular full fare. All of that comes from a discount on the cross-border ticket. Note that this Super Sparpreis Europa fare is non-refundable and good only for the specific Munich to Zurich train you booked. The tickets through Switzerland and France have no discount (and can be changed).

Note that this Super Sparpreis Europa fare is non-refundable and good
only for the specific Munich to Zurich train you booked.

Note however that if you miss that train due to a delay you can of course take the next train at no extra charge.