If I send an email to
Hello@Example.com instead of
firstname.lastname@example.org, could there come up any problems with the delivery because of the uppercase letters?
Technically these are different email addresses, however, you are probably OK.
The domain part of the email address (example.com) is not case-sensitive. So "ExaMple.com" is the same as "example.com". However, the recipient (local-part) is potentially case-sensitive, so "Hello" is different to "hello". Having said that, real world mail servers tend not to enforce this (I have never encountered one that does) - as it is simply confusing and prone to user error.
If you are storing email addresses then you probably should store them in their original case (the recipient at least) to be safe. However, always compare them case-insensitively in order to avoid duplicates.
Most SMTP servers are fine with uppercase characters. However, some do differentiate between upper and lower case characters in the recipient part. The domain part is case insensitive.
For example, google mail does not differentiate between upper and lower case recipients.
I recommend you check with the concerned server admin or leave them a question to query the same.