Suppose a particular route is served by TER trains, as well as Intercités trains which don't require reservation. Is it possible to buy a ticket for that route, valid for a few hours, or a day, or whatever, which can be used on any reservation-free-Intercités or TER train on that route?
TER mean regional trains. They are funded and managed by the region administration and operated by SNCF.
Intercités are national trains operated by SNCF.
The distinction might not be clear, but they are not managed the same way and their pricing is different. This would be weird to be able to pay a certain amount for a TER ticket and get on an Intercités which has a higher ticketed price, for example. You can try to search for a trip on http://www.voyages-sncf.com, you will find fixed-prices for TERs that are different from moving Intercités prices (only Intercités without reservation have fixed prices).
EDIT : As Gilles mentioned, there might not be Intercités without reservation that are priced differently than the TER. But remember that it requires to buy a ticket without a reservation, because any ticket with reservation might have another price.
Full-price (“tarif normal”) open train tickets are valid in all trains taking the route printed on the ticket, regardless of their status, except for trains that require a reservation. They are valid for two months after the travel date indicated on the ticket.
When there are both TER and Intercités or even TGV on the same route, you will usually get the best price if you book a long time in advance, and the worst price if you book at the last minute, with the “tarif normal” valid in trains without compulsory reservation in between.
There are a few ways to get a reduced price on trains without mandatory reservation. Tarif Loisir Week-end gives you a 35% reduction on a return ticket if you spend the night from Saturday to Sunday at your destination and book at least one day in advance; it is only valid for a specific train but this can be a TER where reservations are not possible. Tarif Découverte gives you a 25% reduction at certain times (“période bleue”) if you are under 25, over 60 or traveling with a child.
It's possible that some regions have introduced cheaper tickets valid only in TER, but I'm not aware of any. Many regions do have passes and reduction cards for frequent travelers, some of may only apply in TER.