Try this instead:
procedure TDropbox.Upload(const AFileName: String);
const
API_URL = 'https://api-content.dropbox.com/1/files_put/sandbox/';
var
URL: String;
https: TIdHTTP;
SslIoHandler: TIdSSLIOHandlerSocket;
begin
URL := API_URL+ExtractFileName(AFileName)
+ '?oauth_signature_method=PLAINTEXT&oauth_consumer_key=' + FAppKey
+ '&oauth_token=' + FOAuth.AccessToken
+ '&oauth_signature=' + FAppSecret + '%26' + FOAuth.AccessTokenSecret;
https := TIdHTTP.Create(nil);
try
SslIoHandler := TIdSSLIOHandlerSocket.Create(https);
SslIoHandler.SSLOptions.Method := sslvTLSv1;
SslIoHandler.SSLOptions.Mode := sslmUnassigned;
https.IOHandler := SslIoHandler;
https.Post(URL, AFileName);
finally
FreeAndNil(https);
end;
end;