Is there a downside to encrypting too much data with the same key? Is there a downside to encrypting too much data? If so, what?

I'm guessing things like


too much data encrypted with the same key, possibly leaking information due to a collision
encrypting non-essential data, wasting resources


could be a problem, but that's just a hunch, and that's not how we do cryptography. 

On the other hand, failing to encrypt some data can leak some very sensitive information, and encrypting everything is an easy way to avoid that.
Yes, but the answer is more or less embedded in the question here; you can only say that you encrypt too much data in case the secret key and / or plaintext becomes vulnerable.

Most modes of operation define how much data can be encrypted. This could mean real limits to the amount of data (approx. $2^{36}$ bytes or 64 GiB for AES-GCM) or it may be completely impossible to ever reach the limit (e.g. $2^{128}$ bytes or blocks of data).

When it comes to wasting resources: that's more a question of encrypting the wrong data rather than too much data.

And yes, failing to protect sensitive data is a sin as well. But that's a generic statement that also comes down to categorizing the data correctly.



First categorize the data, then find the right method to encrypt the amount of data that needs confidentiality. Encryption itself is not the goal.