I heard somewhere at one point that only the bishop can hold unopened tithing envelopes during the week – for example, if a member gives a donation to the bishopric after that week's deposit has been prepared, only the bishop can keep that envelope with him until the following Sunday. Donations received by mail are always sent to the bishop's home, which would usually be kept by him during the week until Sunday. But like you, I'm not finding anything specific in the current Handbook.
It's probably a good idea, since the bishop is ultimately accountable and responsible for finances in the ward, but I can't find a policy.
Samuel Bradshaw • If you desire to serve God, you are called to the work.