This station has no DJ or speaking of any kind, just instrumental music. If the Institute is destroyed, Classical Radio will be shut down and cannot be tuned into anymore. However, the pieces can still be heard on any settlement jukebox.
The station's broadcast and the music it plays are the waves used by synths to teleport into and out of the Institute via the molecular relay device.

Johann Sebastian Bach – Suite in E minor, BWV 996 – III. Courante (early 1700s)
Ludwig van Beethoven – String Quartet No. 12 in E-flat major, Op. 127 – I. Maestoso – Allegro (1825)
Johannes Brahms – Tragic Overture in D minor, Op. 81 (1880)
Frédéric Chopin – Ballade No. 1 in G Minor, Op. 23 (1830s)
Frédéric Chopin – Introduction and Polonaise brillante in C major, Op. 3 – I. Introduction. Lento (around 1830)
Frédéric Chopin – Nocturne in E-flat major, Op. 9, No. 2 (1833)
Claude Debussy – La Mer, L. 109, CD. 111 – II. Jeux de vagues (early 1900s)
Claude Debussy – Nocturnes, L. 91, CD. 98 – I. Nuages (1899)
Edward Elgar – Enigma Variations – Variation IX (Adagio) "Nimrod", Op. 36 (late 1890s)
Edward Elgar – Salut d'Amour in E major, Op. 12 (1888)
Edvard Grieg – Peer Gynt, Suite No. 1, Op. 46 – 1. Morning Mood in E major (1875)
Edvard Grieg – Peer Gynt, Suite No. 1, Op. 46 – 4. In the Hall of the Mountain King in B minor (1875)
Gustav Holst – The Planets, Op. 32 – I. Mars, the Bringer of War (1914; condensed variation)
Franz Liszt – Consolations, S.172 – Consolation No. 3 in D-flat major – Lento placido (mid 1800s)
Franz Liszt – Hungarian Rhapsody No. 2 in C-sharp minor, S.244/2 (1847)
Cut content Diego Masson – Valse Diable (1991)
The file exists, Sound\FX\MUS\Radio\Institute\MUS_Institute_Masson_ValseDiable.xwm, but isn't referenced; see YouTube.
Cut content Wolfgang Amadeus Mozart – Requiem in D minor, K. 626 – III. Sequentia – f. Lacrymosa (1791; unfinished)
The sound descriptor (form ID 000ab8d9) points to a non-existing file, Sound\FX\MUS\Radio\Institute\MUS_Institute_Mozart_RequiemDMinorLacrimosa.wav; see YouTube.
Modest Mussorgsky – Méditation (feuillet d'album) (1880)
Modest Mussorgsky – The Fair at Sorochyntsi – Finale: Gopak (late 1800s)
Nikolai Rimsky-Korsakov – Scheherazade, Op. 35 – I. The Sea and Sinbad's Ship (1888)
Camille Saint-Saëns – The Carnival of the Animals – V "L'Éléphant" (The Elephant) (1886)
Camille Saint-Saëns – The Carnival of the Animals – IX "Le Coucou au fond des bois" (The Cuckoo in the Depths of the Woods) (1886)
Camille Saint-Saëns – The Carnival of the Animals – XIII "Le cygne" (The Swan) (1886)
Domenico Scarlatti – Sonata in F minor, K. 19 (1730s)
Robert Schumann – Scenes from Goethe's Faust, WoO 3 – Overture in D minor (mid 1800s)
Bedřich Smetana – 3 Polkas de salon, Op.7 – III. Polka in E major (around 1850)
Johann Strauss II – The Blue Danube, Op. 314 (1866)
Pyotr Ilyich Tchaikovsky – Marche slave in B-flat minor, Op. 31 (1876)
Pyotr Ilyich Tchaikovsky – Swan Lake, Op. 20 – No. 29 Scène finale (mid 1870s)
Richard Wagner – Die Walküre, WWV 86B – Ride of the Valkyries (mid 1800s)

"Valse Diable" was probably scrapped from the game due to the fact that, being that the composer was active during the 20th century, the piece would've been too anachronistic to what the developers were looking for: classical pieces written during the Enlightenment Period and the Romantic Era.
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After the Institute has been destroyed and Classical Radio shut down, the station can be restarted using the following console command on PC: StartQuest RadioInstituteQuest.
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I hope you liked my guide, thanks for reading it :). Have a good day!

Information taken via https://fallout.fandom.com/wiki/Classical_Radio_(Fallout_4).