Can a Trustee of an Irrevocable Trust request a withdrawal of funds be issued to his name only, or should the check be able to the individual as Trustee of the John Doe Irrevocable Trust?

I assume when you established the account you verified the powers of the trustee. If they have the power to withdraw funds, who those funds are payable to is not really your concern.

And it is also very possible that the trust includes the trustee as a beneficiary, too. I second Randy's opinion.

On the other side of the issue, if the trustee were to bring in a check payable to him as trustee or to the trust, it would be wrong to allow him to deposit it to an account he owns personally. That would put the bank on notice of a breach of fiduciary trust under UCC 3-307. The check should be deposited to an account in the trust's name. What happens from there we have discussed here already.

Thank you both very much. In this instance the Trustee is not the beneficiary.