Enterprise backup schedule

Once you jump into the mid range TB's tapes become a hassle. You can go LTO4 which will increase your speed a good bit and double your storage per tape ratio, not to mention giving hardware based encryption potential. LTO4 drives are very pricey though. However, tape pricing has been falling like gasoline as of late thankfully! But it sounds like your issue is mainly having enough hours in the backup window to complete the job. You could consider stopping tape backups, and use WAN or Internet based backup solutions. An ideal world would involve your main storage farm replicating offsite and abandoning tapes, and using snapshot technology for your incremental file change restores.

We have 24 TB of weekly backups that we try to get off-site (iron mountain). Is that the norm? What are the disadvantages of monthly full backup instead of weekly?

I like the idea of off-site backups over WAN. The problem is we will need 1 gb access link to transfer 24 TB over a weekend.

There are very good block level backup solutions out there to help with the backup window for incrementals. Iron Mountains flavor of this is Delta Block. Block level solutions help with space consumption and speed.
Moving from weekly to monthly fulls is more of a compliance/risk management dept. question if they are comfortable with this. There are no real disadvantages or advantages from moving to monthly fulls.
Also having good WAN acceleration appliances like Riverbed can help provide proper allocation of your WAN resources to speed backups along.

I am trying to understand backups, if I do monthly full and daily incremental I will need all the incremental to a specific date to restore? The main advantage of weekly full is you will need less incremental backups to do a restore?