A few questions regarding a Revocable Trust which becomes Irrevocable upon the death of the Grantor/Trustee.

1. Should the Revocable Trust be closed and a new account opened for the Irrevocable Trust?
2. If a new Irrevocable Trust account is opened, can checks payable to the Revocable Trust be deposited into that account?

The trust changes its nature, but not its name (if revocable is part of the name, that does not change,either). There is no reason to change accounts. You will want to obtain documentation of the new trustee(s). They may or may not have been named as successor trustees in the trust document. You will also require the trust to obtain an EIN of its own. You can no longer use the SSN of a grantor.

Because the account title isn't going to change, your second question is moot.