You can usually run in a newer version without problems as long as the bitness of the Office instalation is the same. If your installation of OFFICE is 32 bit and you are distributing as an .accde, then the run version of Office must also be 32 bit. Most Windows installations are 64 bit but up until very recently 32 bit was the default Office install. If you are distributing an .accdb, 32 bit should run in 64 bit but not vice-versa.
Running a db in an older version that that which created it is problematic if you need any references to other Office products. Access will upgrade references but it will not downgrade them. You should always develop in the lowest version your user base will use to avoid reference issues. If you use a newer version, then you need to use early binding so that the references are not resolved until runtime.