What happens when a TPM chip breaks or fails?
The purpose of a TPM module is to ensure that there is absolutely no way to obtain the keys stored on it. Whether or not it actually fulfills that purpose is a topic for another question. For now let's assume that it works as designed. That means when the TPM module is destroyed, so is the key, and so is any hope to decrypt the data encrypted with it.
However, most backup solutions do not binary mirror the hard drive they back up. A proper backup solution must provide the ability to fully recover the data even in case the original system is a complete physical loss. So any dependence on any part of the backed up system would be counter-productive. Usually a backup solution backs up the cleartext data. There are also solutions where the backup is encrypted with a key before sending it to a backup system but in that case the key should be backed up elsewhere (this means you need two backup systems: A small-space/high-security one for the key and a large-space/low-security one for the data). The backup system might or might not encrypt the backups independently with its own key on its own hard drives.
If the TPM 'breaks' or becomes otherwise inaccessible, all cryptography dependent on keys stored by the TPM also breaks. This is the most concise and inclusive way I could think to put it.
I say 'cryptography' because the TPM does more than just encryption. Encryption is just one use of cryptography, as are signatures, authentication, etc.
The TPM stores keys, it doesn't actually do the drive encryption. It acts as the key-ring for the software that actually does the encryption.
Think of losing your physical keys: if you asked "what happens when I lose my keys", we can't answer that for you. We don't know what keys you have, or what they are for. All we can say is "well, whatever you had keys to on your keys, now you can't access those things."
What exactly happens is not a general question about keys at all. If you had keys to your house, a car and a boat on your key ring, and you lost it, but you have a backup house key somewhere, then the answer is you lose access to your car and your boat, but you can still get in your house. Sorry, but: duh!
So back to your encrypted hard disk: What exactly happens is not a TPM question at all, it's a question about the software you are using. If you are using BitLocker, for example, to encrypt your hark disk, you might know everything there is to know about the TPM, but if you don't know exactly how BitLocker works, and how/if it even uses the TPM, then you cannot answer the question of what 'exactly' happens. Does it migrate/backup keys? Does it save clear-text data to the cloud somewhere for retrieval? These are questions for the BitLocker software, and not about the TPM at all.
So in short, if the TPM 'breaks':
- Any data you encrypted with a key that only exists in the TPM, which isn't backed up, is lost (i.e. your encrypted hard disk)
- Any cryptographic identity based on the TPM (i.e. Identity Keys) is now lost
- Any trust in the platform (i.e. during remote attestation) is now lost
IOW, any cryptography based on the TPM is now hosed.
NOTE: As in another answer "broken" here means "inaccessible", not "hacked" or broken in that it becomes insecure, that is another question.