I am traveling by train from Nice, France to Moscow, which goes through Belarus but I am not disembarking anywhere along the route.
I have a Russian visa. Do I need a transit visa for Belarus?
Basically, it is quite possible that you just cannot go at all. Russia and Belarus have had a Union State for some 20 years, with no border controls between them. All went well for years; third-country nationals filled a common Russo-Belarussian migration card and were admitted at Belarussian border and then proceeded to Russia without any delay or hindrance.
However, in October 2016 Russian government made a declaration that according to its view of the Union State treaty, simplified border crossing means that all border crossing points between Belarus and Russia are bilateral instead of multilateral and only citizens of the Union State may cross it legally. A number of Ukrainians and Poles who crossed into Russia through Belarus were detained, tried for unlawful presence and then expelled. Lots of Polish trucks were turned back on the main road connecting Warsaw, Minsk and Moscow.
So, technically, according to the Russian govt's current interpretation of Russo-Belarussian border setup, no third-country national may legally cross that border, be that by train, car, bus or plane.
Yes you do. It is only when transiting airports without entering the country that you don't need a visa. With a few exceptions, when entering any country by land, you need at least a transit visa.
This is also stated on the GOV.UK website: you need a Belarusian transit visa.
Furthermore, one should bear in mind that Belarus and Russia (which share a common border) have introduced a restriction whereby only citizens of these countries can cross the border. However, this is currently only enforced on the roads, and not when travelling by rail.
You don't need a transit visa anymore since 2020-07-01.
As of 2020-07-01, Russia and Belarus recognise each others visas. That means you can use your Russian visa all the way to the Belarus-Poland (or -Latvia, -Lithuania, -Ukraine) border. See also this answer about the Belarus-Russia border.
Warning as of 2020-07-07: another source in Russian, dated 2020-07-01, states it will take several more months until it actually takes force. These sources contradict each other. Either it's already in force, or it will be soon (and as of July 2020, the 2019-2021 Coronavirus Pandemic means few travellers should make the overland EU-Belarus-Russia trip anyway).