You can copy a filemaker DB to you iPhone/iPad
1.) using iTunes:
(scroll down to see the bottom of the Apps Tab while your device is mounted) from there you can drag the file, or located it and copy it over via open/save dialog box
2.) email it yourself,acess it via "Open with" from Mail when opening attachment (I HAVEN't Tried this, so can't give specifics)
I'm not sure if copying over to dropbox, or using GoodReader - again using Open with services would work.. perhaps someone else will pipe in.