Trouble backing up remote database from local server
As is the command is trying to execute on your LOCAL instance and not REMOTE. You need to have it execute through the linked server itself. You can build the query as a dynamic statement and then simply execute the query against sp_executesql on the linked server. So something like this should:

DECLARE @qry varchar(2000)
SET @qry = 'BACKUP DATABASE [remotedb] TO  DISK = N''C:\Temp\remotedb.bak'''
EXEC REMOTE.master.dbo.sp_executesql @qry