How do I download a file or photo that was sent to my Telegram bot?

This is now available!

https://core.telegram.org/bots/api#getfile

Hooray! It was added on Sep 18th:

https://core.telegram.org/bots/api

Usage:

In the JSON of the message you will receive a file_id as before. An example of a message object with a voice file:

{
  message_id: 2675,
  from: {
    id: 10000001,
    first_name: 'john',
    username: 'john'
  },
  chat: {
    id: 10000001,
    first_name: 'john',
    username: 'john'
  },
  date: 1442848171,
  voice: {
    duration: 2,
    mime_type: 'audio/ogg',
    file_id: 'AwADBAADYwADO1wlBuF1ogMa7HnMAg',  //  <------- file_id
    file_size: 17746
  }
}

Via the API's getFile you can now get the required path information for the file:

https://api.telegram.org/bot<bot_token>/getFile?file_id=the_file_id

This will return an object with file_id, file_size and file_path. You can then use the file_path to download the file:

https://api.telegram.org/file/bot<token>/<file_path>

Note that this link will only be available for an hour. After an hour you can request another link. This means that if you want to host the file somehow and you rather avoid checking and re-checking for fresh links every time you serve it you might be better off downloading the file to your own hosting service.

The maximum size of a file obtained through this method is 20MB. Error: Obtained when a file large than 20mb is used.(Shown below)

{"ok":false,"error_code":400,"description":"Bad Request: file is too big[size:1556925644]"}

From telegram's docs:

On success, a File object is returned. The file can then be downloaded via the link https://api.telegram.org/file/bot/, where is taken from the response. It is guaranteed that the link will be valid for at least 1 hour. When the link expires, a new one can be requested by calling getFile again.For the moment, bots can download files of up to 20MB in size.

Yay! It's just added at September 18, 2015

You can use getFile(file_id). This function returns a File object containing file_path. You can download the file through this address:

https://api.telegram.org/file/bot<token>/<file_path>

As mentioned in Telegram Bot API Documentation, the File object will be valid for about one hour. You should call getFile again to get a new File object if the old one expires.