If it were, for example, someone trying to evade forum bans or other such measures by pretending to be someone else which has happened in the past, both accounts would be banned from the entire forum upon discovery and most likely so would the IP. Circumventing staff actions carries deadly consequences for the accounts in question.
If it were two people just trying to share a connection there'd be no problems at all.
I'm sure you're not planning to do anything like the former, but since you brought this topic to the table I might as well answer it thoroughly for anyone else who might be thinking about this kind of thing in the future.
We're very good at spotting people trying to be deceptive, so anyone thinking of evading bans and whatnot with second accounts will almost certainly be caught out. There are also certain factors that can determine whether two accounts are really on different computers or just on one.