Most of Toynbee’s papers are stored in the Western Manuscripts division of the Bodleian Library in Oxford.
The National Register of Archives (accessed 2011 August 20) also mentions
Correspondence with Sir Herbert Butterfield from 1950 to 1969 in the Department of Manuscripts and University Archives, Cambridge University Library
Letters to Rosalind, Countess of Carlisle in the archives at Castle Howard
Correspondence with Lord Cherwell from 1920 to 1933 in Nuffield College Library, Oxford
Correspondence with Lionel Curtis from 1931 to 1954 in the Western Manuscripts division of the Bodleian Library
Correspondence with Philip Kerr from 1927 to 1929 in the National Records of Scotland, Edinburgh
Correspondence with Sir BH Liddell Hart from 1936 to 1968 at the Liddell Hart Centre for Military Archives, King’s College, London
Correspondence with Gilbert Murray from 1915 to 1958 in the Western Manuscripts division of the Bodleian Library
Letters to Herbert Samuel from 1935 to 1937 in the Parliamentary Archives and
Letters to Sir Alfred Zimmern from 1909 to 1957 in the Western Manuscripts division of the Bodleian Library.
Scattered documents in various places in The National Archives:
Balliol College, Oxford and Chatham House hold some papers and correspondence.
The Edward C Carter papers at the Columbia University Libraries Archival Collections contain correspondence with Toynbee. That may mean letters to and from, as it appears to mean in the National Register list.
The School of Slavonic and East European Studies Library, University College London, holds RW Seton-Watson’s correspondence with Toynbee from 1917 to 1951. The Seton-Watson papers also contain material on the Foreign Research and Press Service, which Toynbee ran from 1939 to 1943.
The Archives of the LSE Library hold Martin Wight’s correspondence with Toynbee.
There will be material in other collections.
It isn’t clear whether some of the institutions William McNeill thanks in his Arnold J. Toynbee, A Life, New York, OUP, 1989 held papers or merely provided him with a base for general research, but he mentions the archives of Time, Inc in New York and writings by Toynbee lodged at the Nihon University Library in Tokyo.
I refer to sound and film archive material here.