Can you send heirlooms to another server?

The "band-aid fix" right now is through the mail system as Whylune explained it. However, in WoD they're going to make Heirlooms have their own interface - reducing the bag space they take up, while being accessible to all of your characters on that account.

The mail thing is just to tie us over til then :P

There's other account-bound items that can benefit from account-wide mail. Like the Sonic Pulse Generator that I found from archeology on my Druid, faction insignias dropped by Zandalari Warscouts and Warbringers, etc..