In January 2004 Putin and Kazakh president Nursultan Nazarbayev reached an agreement on extending the lease of Baikonur to 2050. Therefore the official stance is that Baikonur will continue to exist as a launch base until at least 2050. Roskosmos chief Igor Komarov reiterated this this last June (see this TASS report, only in Russian) :
http://tass.ru/kosmos/2021495He said that piloted flights from Baikonur will continue until at least 2024, but at the same time added that "piloted flights, commercial missions and all other missions" will be gradually transferred to Vostochnyy. Therefore one does wonder how legally binding the lease agreement is. It's hard to imagine the Russians continuing to pay Kazakhstan $115 million a year for the use of Baikonur if they're going to shift all their launches to Vostochnyy long before 2050.
So what remains in the offing for Baikonur? Piloted Soyuz launches until 2024 or possibly longer if ISS operations are extended or if Russia splits off part of its segment from the ISS to operate as an independent station (and assuming the Soyuz successor, which will fly on Angara from Vostochnyy, is further delayed, which is entirely plausible). Unmanned Soyuz launches could theoretically all be transferred to Vostochnyy within a matter of years.
Proton is expected to be phased out in favour of Angara by 2025. Other than that, there is the joint Russian-Kazakh Baiterek venture, which originally planned to use the Angara-5, then switched to Zenit and now finds itself without a launch vehicle, even though new proposals for Baiterek are expected to be formulated before 25 January according to this TASS report (in Russian) :
http://tass.ru/kosmos/2562543All in all, despite all the official statements, Baikonur's future is looking pretty bleak...