That depends on the country and train company. In Switzerland you can upgrade and in Sweden you can't. 

What specific route do you want to upgrade?

can I upgrade the ticket class in France?

For domestic journeys in France, you can.

Be aware the upgrade fee is not as simple as reserving a first class seat with a second class pass.

You need to check and, if available buy, with each operator before boarding and often the upgrade will be calculated as the difference between the standard first and second class fares, even for pass holders.

If you do not clear this before boarding the train you may be faced with significant penalties if in first class with a second class pass.

In France, though, it kind of works that way: you indicate you have a 2nd class pass and can then select 1st class seats - at a higher price than reservations for a 1st class pass. You can do that on the b-europe page:

https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish

IIRC it also works like that on the SNCF ticket machines.

Thanks for the info - a trial shows that a first class resv’n with a second class pass can indeed be bought - Lille to Paris added 9 euros and Lille to Marseilles (direct) added 30 euros to the 10 euro normal fee.