Does encrypting twice using the same block cipher produce a security weakness? If I use the output of a cipher, for example a block cipher such as AES and encrypt it again with the same algorithm, I read that this introduces weaknesses into the overall security of the system.

Is this the case?
Well, I don't know where you read that.  However, depending on what they meant by that, that's either nonsense, or an appropriate cautionary tale.

What is known is that encrypting again cannot make the cipher weaker, as long as the keys you chose to encrypt a second time are independent of the keys you use the first time.  The logic behind this is straight-forward; the attacker is assumed to know everything about the cipher except the keys.  If he has some ciphertext that he wants to attack, there's nothing preventing him from picking his own "second time key", and then encrypting the ciphertext to form a doubly-encrypted ciphertext.  If double-encrypting was inherently weaker, then he could attack that (and hence single-encryption would be equally weak).

On the other hand, if the two keys you use during the first and the second use are the same, this logic does not apply.  And, in fact, there are ciphers where this encrypting twice with the same key is a Bad Idea.  For example, some ciphers will, if you run the ciphertext through a second time with the same key, will generate a 'doubly-encrypted ciphertext' that is exactly the same as the plaintext; this is obviously weaker than the 'singly-encrypted ciphertext'