Instead of using a visual basic macro which will have to be run every time you change either of the lists (A or B), you can use Conditional formatting.
You have two groups of names - 'existing', which you check against and 'new' which have to be checked against the 'existing' list.
The new names are organized with first names in column B and last names in column C
1. Select the column of first names in the 'existing' list. Name this selection "FirstNamesDB"
To name a range, first select it, then Insert - Name -Define. The address range should already be in the box at the bottom. Enter the name at the top and click Add, then OK
2. Repeat for the column of last names in the 'existing' list. Name this selection "LastNamesDB"
3. Select the whole table of 'existing' names (2 columns). Name this selection "AllNamesDB"
4. Go to the first cell containing a new name (First name).
For this example the table of new names starts on row 23, so the first new 'first name' is in cell B23.
4a. Select Format - Conditional Formatting...
4b. Under 'Condition 1' select 'Formula Is' from the drop-down list
4c. In the formula box enter the following:
Note 1: This is all one line, to go in the formula box
Note 2: Make sure the $ signs are used as shown - this allows the conditional format to be copied to the other cells and maintain the correct cell references.
Note 3: TRUE and FALSE do not have quotes around them - they are logical states.
Note 4: The names you gave to the 'existing names' ranges do not have quotes
4d. Select the format you want to show when a First name/Last name pair does not exist in the existing names list.
4e. click OK
5. Select the cell you have just added the conditional formatting to, and Copy it
6. Select all the cells in the new list (two columns) and use Paste Special - Paste Formats.
7. When a name pair does not exist in the existing names list, the first and last names will show your selected format.
Basically this works by searching for both first and last name in the two columns of existing names using Vlookup. If either name is not present Vlookup gives an error and returns TRUE. If both names occur in their respective columns then the final Vlookup checks that they are both on the same row. If they are present but not on the same row, it returns TRUE. When the formula returns TRUE it triggers the conditional formatting.