Short answer: yes and no.
Using three accounts won't run the battery down in itself: by far the bigger factor is how much activity there is. To take an example, Gmail uses push messages (via Google Cloud Messaging, GCM) to notify your phone of new mail, so the phone doesn't have to repeatedly poll the server for new messages for each account.
Because of this, adding a new account that never gets any email will have hardly any effect on battery use. OTOH, doubling the amount of email you get will have a much bigger effect, because GCM is waking up your phone twice as often. While adding a new account makes little difference on its own, you're probably adding the account because it's going to get mail, so you will end up using the battery more.
The same applies to other services too. Even for anything that does poll instead of using GCM, Android is clever enough to do all the polling tasks at the same time, to avoid repeatedly waking up your phone. This means that checking two accounts uses a little more battery than checking one, but less than twice as much: again, the amount of work it's doing each update (how much data it's fetching) is a bigger factor.
To address your particular situation, I'd guess that the background updates on your phone aren't using nearly as much battery as the foreground things you're doing. Don't forget you check the battery use of each app in the device settings.
