They stopped running in 2015. There's a good article here
https://www.connaissance-du-rail.com/reportages-et-photos/la-fin-des-tgv-postaux/ (apologies it's in French - I've added a google translate version below)
After more than 30 years of good and loyal service, postal TGVs▸ finally bowed out on Saturday June 27, 2015, ending more than a century and a half of collaboration between the postal administration and the railway.
Postal TGV Paris - Cavaillon. Rame 951. Vinneuf
On June 8, 2015, the 951 train ensures the TGV 6997 Paris - Cavaillon, seen here in Vinneuf, in Yonne. Photo Pierre BAZIN
It was in 1982 that La Poste, attracted by the TGV formula, decided to acquire this type of equipment for its rapid delivery.
Two trains are then ordered.
Composed of two drive units framing eight trailers, each with a large side door on each side, they can take 250 specific containers loaded with mail and parcels.
In fact, five half-trains, numbered 901 to 905, called sections, will be delivered to ensure the reserve.
Assigned to the Paris-Sud-Est depot, they will enter into service from October 1, 1984.
Each day, they will make two round trips between Paris-Charolais and Lyon-Montrochet, where a warehouse has been specially fitted out. One of the two relationships marks a stop at Mâcon-Loché, where a specific platform has also been created on the service connection joining the high-speed line to the conventional line.
Postal TGV. Rame 952 in Marolels-sur-Seine. Train 6997 Paris - Cavaillon
Train 952 in Marolles-sur-Seine, June 16, 2015, with the TGV 6997 Paris - Cavaillon. Photo Pierre BAZIN
On November 7, 1994, a new sorting center was put into service in Cavaillon.
As a result, Lyon-Montrochet was abandoned and the new service now consists of four daily connections from the capital: two to Cavaillon and two to Mâcon.
This requires the use of an additional train obtained by transforming the first class South-East train no. 38, which will become the postal sections no. 906 and 907.
Postal TGV to Crisenoy. Rame 951. TGV 6991 Paris - Mâcon
Passage to Crisenoy of the 951 train with the TGV 6991 Paris - Mâcon. Photo Pierre BAZIN, June 17, 2015
In 2003, the operating numbers of the trains changed according to the following scheme:
- Train 951: drive units 923-001 and 923-006 (ex sections 901 and 906)
- Rame 952: drive units 923-003 and 923-005 (ex-sections 903 and 905)
- Rame 953: drive units 923-004 and 923-007 (ex sections 904 and 907)
- Train 954: power train 923-002 (ex-section 902); half-oar ensuring the reserve.
In 2006, a new livery, with much more visible Post logos, appeared.
At the same time, the service of the Atlantic facade is mentioned. But the idea was quickly abandoned due to the fall in the number of priority letters to be sent, a phenomenon accentuated by the development of electronic messaging and the success of the "green letter", delivered in two days.
Postal TGV to Champdeuil. Rame 953. TGV 6996 Cavaillon - Paris
In the early morning at Champdeuil, on the high-speed Interconnection line, the 953 train goes to Paris with the TGV 6996 from Cavaillon. Photo Pierre BAZIN. June 26, 2015
As the volume of mail transported continues to decrease, the round trip ¨Paris - Cavaillon de jour was eliminated in 2009.
Since then, only three daily rotations remained, one on Cavaillon, two on Mâcon, which enabled postal trains to make average monthly journeys of 17,600 km, a total of approximately 8,500,000 km since their implementation service in 1984.
But the fall in the volumes of mail transported continuing, it was decided to definitively stop the service of postal TGV on Saturday June 27, 2015.
This led to the closure of the Paris-Charolais sorting center, replaced by a new warehouse in Bonneuil-sur-Marne, grouping mail to the entire territory.
In the future, it is planned to send mail by swap bodies, either by conventional freight trains or by trucks, depending on the destinations. However, nothing has yet been decided on the future of rail transport in the area of mail transport.
La Poste would like to resell the trains, but given their special architecture, they are more likely to be scrapped.