We have agreed to cash payroll checks for a construction company.
The payee of these payroll checks are not from the USA. We are running them through OFAC before cashing the check but do we need to continue checking the list each time the OFAC list updates?

Yes, absolutely.

Also I am saying that we need to see work VISA and not just their countries passport or country ID's. I'm I wrong in requiring their work VISA as proof of Identity?

For how long do we need to check the OFAC list after the non-customer cashes his/her check?

A valid (non-expired) passport should be fine. Whether the person is eligible for U.S. employment should be left up to the USCIS or the IRS' discretion. Don't do anything that you're not required to do. As for checking against the OFAC list, it is a risk-based decision. However, best practice for this particular individual would be running a scan on every single transaction.

Our tellers run the OFAC check. After we (BSA Dept.) have reviewed the "false hit" report that comes from our OFAC interdict software, we will add the the person to our "previously approved false-hit list". The tellers will still have to perform OFAC screening, but won't have to contact the BSA Dept. for review and approval, as long as the information entered and the false hit output are identical to the information that was previously approved. The tellers just have to refer to the "previously approved" list, which speeds up the process of cashing the check.

I've heard ICE agents describe a VISA as the equivalent of a hall pass; i.e. it indicates the person has a right to be here, it is not evidence of identity. However, since you are cashing payroll checks, there is some logic in requiring it as a form of secondary identification with a passport.

If you will be cashing checks on a recurring basis for primarily the same individuals, you might want to consider creating a CIF record for them in your core system. You could put a message indicating this is a check-cashing customer of ABC Company (if your system will do this) and set the record retention period accordingly. In most systems I am familiar with, this would enable your system to automatically perform OFAC checks on an ongoing basis.