Although the accepted answer only shortly addresses it, I'd like to provide more detail.
GNSS: Global Navigation Satellite System. This is a satellite constellation that provides a global positioning, navigation and timing (PNT) service. As of 2015 we have GPS (USA), GLONASS (Russia) and in development are Galileo (Europe) and BDS (China, also referred to as BeiDou).
RNSS: Regional Navigation Satellite System. Similar to a GNSS (it is also a satellite constellation) but the satellites only allow for regional positioning (often on top of what GNSS already provides). As of 2015 we have BeiDou (China), and in development are IRNSS (India) and QZSS (Japan).
SBAS: Satellite Based Augmentation System. These are geostationary satellites that submit augmentation signals that provide integrity services to GNSS, and allow for some accuracy improvement. As of 2015 we have WAAS (USA), EGNOS (Europe), MSAS (Japan), GAGAN (India) and SDCM (Russia). Except for SDCM, which augments GLONASS and GPS, all others only augment GPS.
DGNSS: Differential-GNSS. This is a technique by which one or more reference base stations are used to provide corrections to a measurement station so that high accuracy positioning can be obtained. DGPS is often used when you only use the GPS constellation. RTK is real-time kinematic, in which a communication channel is present so you get real-time result (compared to non-RTK methods which only focus on post-processing of the data). A continuous operating reference station (CORS) network is a DGNSS technique.
PPP: Precise-Point Positioning. Although you can obtain similar accuracies as DGNSS, it is not a DGNSS technique as it does not use one or more reference base stations. Instead, it uses correction messages from (for example) services available on the internet (developed for example by the International GNSS Service, IGS).
This is of course just a short summary and many details are omitted. If you like to know more, there is plenty of information online. A good source of information is Navipedia.
The two answers pretty much cover it and are not really contradictory. Shutting down those stations will mean no broadcast correction signal will be available to recievers tuned to the US GPS network in that area. The other GPS systems don't use that same network - they have their own ground stations. And as Mapperz points out, there are different kinds of ground stations. The ones in Russia for the US system are for recievers (user segment), not for network and satellite maintenance (control segment). See the Wikipedia entry on the US system for details.
Nothing will happen to the US GPS network itself. Accuracy will get worse in those areas (and only those areas) covered by those ground stations shut down because a broadcast correction signal will not be available. How much is already answered - back to 3m from potential 10cm.
In theory they want to promote use of their own GPS system, and the same correction functionality in the US system has a counterpart in the GLONASS system. Recievers just have to be able to use the GLONASS ground control/correction station signal. As I recall, there is a class of units manufactured to use either or multiple systems. You're basically looking at an equipment upgrade/change for anyone who was relying solely on the US GPS system in Russia.
Best Answer
Here's a link to the CORS map.
And here's a link to a KMZ file of CORS sites. It doesn't look like India has a CORS network.
It looks like most of the CORS locations are in the United States.
Here's the website on other GPS Augmentations.